What is a Quadratic Equation? What is a Quadratic?

A quadratic equation is a second-degree polynomial equation in a single variable x. This means the highest power of the variable x is 2.

The standard form of a quadratic equation is:

Where:

• x is the variable.
• a, b, and c are coefficients, which are known numbers.
• Crucially, a cannot be equal to 0 (if a=0, the equation becomes linear, not quadratic).

The term "quadratic" itself comes from the Latin word "quadratus," meaning square, referring to the x2 term. "What is a quadratic" generally refers to either a quadratic equation or a quadratic function/expression.

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What is the Quadratic Formula? What's the Quadratic Formula?

The quadratic formula is a formula that provides the solution(s) to any quadratic equation in the standard form ax2 + bx + c = 0.

The formula is:

This formula gives two possible values for x (due to the ± symbol), which are the roots or solutions of the quadratic equation.

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What is a, b, and c in the Quadratic Formula? What is 'a' in a Quadratic Equation/Function?

In the standard form of a quadratic equation, ax2 + bx + c = 0, and consequently in the quadratic formula:

• a is the coefficient of the x2 term (the quadratic term).
• b is the coefficient of the x term (the linear term).
• c is the constant term (the term without x).

It's essential to correctly identify a, b, and c from a given quadratic equation before plugging them into the formula. Remember that a ≠ 0.

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How to Solve Quadratic Equations? How to Do Quadratics/Quadratic Equations?

There are several methods to solve quadratic equations (i.e., find the values of x that satisfy the equation):

1. Factoring:
   • Rewrite the quadratic equation in standard form (ax2 + bx + c = 0).
   • Factor the quadratic expression on the left side into two linear factors.
   • Set each linear factor equal to zero and solve for x. This method is often quickest if the quadratic expression is easily factorable.
2. Using the Quadratic Formula:
   • This method works for all quadratic equations.
   • Identify a, b, and c from the standard form.
   • Substitute these values into the formula: x = [-b ± √(b2 - 4ac)] / 2a and calculate the value(s) of x.
3. Completing the Square:
   • This method involves manipulating the equation algebraically to create a perfect square trinomial on one side, which can then be solved by taking the square root of both sides.
   • It's a fundamental method that also leads to the derivation of the quadratic formula itself.
4. Graphing:
   • Graph the corresponding quadratic function y = ax2 + bx + c.
   • The x-intercepts (where the graph crosses the x-axis, i.e., where y=0) are the real solutions to the quadratic equation ax2 + bx + c = 0. This method is good for visualization and finding approximate solutions but may not be precise for exact values.

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How to Use the Quadratic Formula? How to Do/Calculate Quadratic Formula?

To use the quadratic formula x = [-b ± √(b2 - 4ac)] / 2a:

1. Write the equation in standard form: Ensure your quadratic equation is arranged as ax2 + bx + c = 0.
2. Identify coefficients: Determine the values of a (coefficient of x2), b (coefficient of x), and c (the constant term). Pay attention to signs.
3. Substitute into the formula: Carefully plug the values of a, b, and c into the quadratic formula.
4. Calculate the discriminant: First, calculate the value inside the square root: D = b2 - 4ac. This is called the discriminant.
5. Calculate the square root: Find the square root of the discriminant, √D.
   • If D > 0, there are two distinct real solutions.
   • If D = 0, there is one real solution (a repeated root).
   • If D < 0, there are two complex conjugate solutions (no real solutions).
6. Solve for x: Calculate the two possible values for x using the ± sign:
   • x1 = (-b + √D) / 2a
   • x2 = (-b - √D) / 2a
7. Simplify: Simplify your answers if possible.

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When to Use the Quadratic Formula? What is the Quadratic Formula Used For? What Does it Do?

The quadratic formula is used to find the roots (or solutions) of any quadratic equation of the form ax2 + bx + c = 0.

When to use it:

• When factoring the quadratic expression is difficult or not immediately obvious.
• When you need to find exact solutions, including irrational or complex roots, which factoring might not easily reveal.
• It is a universal method that always works, unlike factoring, which is only straightforward for certain types of quadratics.
• It's also used in the derivation of other mathematical concepts and in various applications in physics, engineering, and economics where quadratic relationships arise.

"What does the quadratic formula do?" It provides a direct method to calculate the values of x that make the quadratic equation true.

How to Factor Quadratic Equations? How to Factor Quadratics?

Factoring a quadratic equation ax2 + bx + c = 0 involves rewriting the quadratic expression as a product of two linear factors.

Steps (for a=1, i.e., x2 + bx + c):

1. Find two numbers that multiply to give c and add to give b. Let these numbers be p and q.
2. The factored form will be (x + p)(x + q) = 0.
3. Set each factor to zero to find the solutions: x + p = 0 => x = -p, and x + q = 0 => x = -q.

Steps (for a ≠ 1, i.e., ax2 + bx + c - often called the "ac method" or grouping):

1. Find two numbers that multiply to give a × c and add to give b. Let these be p and q.
2. Rewrite the middle term bx as px + qx: ax2 + px + qx + c = 0.
3. Factor by grouping: Group the first two terms and the last two terms, and factor out the greatest common factor (GCF) from each group. x(ax + p) + y(ax + p) = 0 (where y is the GCF of qx and c)
4. Factor out the common binomial factor: (ax + p)(x + y) = 0.
5. Set each factor to zero and solve.

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What is a Quadratic Function? How to Graph Quadratic Functions/Quadratics/Equations?

A quadratic function is a function that can be written in the form:

or

where a, b, c, h, and k are constants, and a ≠ 0.

The graph of a quadratic function is a U-shaped curve called a parabola. "A graph of a quadratic function" would show this shape.

Example of a parabola (Image: Wikimedia Commons)

Key features to graph a quadratic function (f(x) = ax2 + bx + c):

1. Direction of Opening:
   • If a > 0, the parabola opens upwards (like a smile 😊).
   • If a < 0, the parabola opens downwards (like a frown ☹️).
2. Vertex: The highest or lowest point of the parabola.
   • The x-coordinate of the vertex is h = -b / (2a).
   • The y-coordinate of the vertex is k = f(h) (substitute h back into the function).
   • In vertex form f(x) = a(x - h)2 + k, the vertex is (h, k).
3. Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = -b / (2a) (or x = h).
4. Y-intercept: The point where the parabola crosses the y-axis. Set x = 0 in the function: f(0) = c. So, the y-intercept is (0, c).
5. X-intercepts (Roots/Zeros): The points where the parabola crosses the x-axis. Set f(x) = 0 and solve the quadratic equation ax2 + bx + c = 0 (using factoring, quadratic formula, etc.). There can be two, one, or no real x-intercepts.
6. Additional Points: Plot a few more points by choosing x-values on either side of the axis of symmetry and finding the corresponding f(x) values. Use symmetry to find their mirror points.

Plot these key features and connect them with a smooth curve to graph the parabola.

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How to Find the Vertex of a Quadratic Function/Equation?

The vertex is the turning point of the parabola (the minimum point if a > 0, or the maximum point if a < 0).

For a quadratic function in standard form f(x) = ax2 + bx + c:

• The x-coordinate of the vertex (h) is given by the formula: h = -b / (2a).
• To find the y-coordinate of the vertex (k), substitute this h value back into the original function: k = f(-b / (2a)).
• The vertex is the point (h, k).

For a quadratic function in vertex form f(x) = a(x - h)2 + k:

• The vertex is directly identifiable as the point (h, k). (Be careful with the sign of h; if it's (x+h)2, then the x-coordinate is -h).

When asked to find the vertex of a "quadratic equation," it usually implies finding the vertex of the corresponding quadratic function y = ax2 + bx + c.

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What is the Discriminant of a Quadratic Equation? What is Discriminant in Quadratic Formula?

The discriminant of a quadratic equation ax2 + bx + c = 0 is the part of the quadratic formula under the square root sign.

The discriminant tells us about the nature and number of the roots (solutions) of the quadratic equation without actually solving for them:

• If D > 0 (positive): The equation has two distinct real roots. The parabola intersects the x-axis at two different points.
• If D = 0 (zero): The equation has one real root (a repeated root or a double root). The vertex of the parabola is on the x-axis.
• If D < 0 (negative): The equation has no real roots. It has two complex conjugate roots. The parabola does not intersect the x-axis.

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What are the Roots of a Quadratic Equation? How to Find the Roots?

The roots of a quadratic equation ax2 + bx + c = 0 are the values of x that make the equation true. They are also known as the solutions or zeros of the equation (or the x-intercepts of the corresponding quadratic function's graph).

How to find the roots:

• Use the quadratic formula: x = [-b ± √(b2 - 4ac)] / 2a.
• Factor the quadratic expression and set each factor to zero.
• Complete the square.

The discriminant (b2 - 4ac) determines how many and what type of roots the equation has (two distinct real, one repeated real, or two complex conjugate roots).

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How to Find X-intercepts / Y-intercept of a Quadratic Function?

• X-intercepts: These are the points where the graph of f(x) = ax2 + bx + c crosses the x-axis. At these points, f(x) = 0. So, to find the x-intercepts, you solve the quadratic equation ax2 + bx + c = 0. The solutions are the x-coordinates of the x-intercepts. (See "How to Find the Roots").
• Y-intercept: This is the point where the graph crosses the y-axis. At this point, x = 0. To find the y-intercept, substitute x = 0 into the function: f(0) = a(0)2 + b(0) + c = c. So, the y-intercept is the point (0, c).

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How to Find the Range of a Quadratic Function?

The range of a function is the set of all possible output values (f(x) or y values).

For a quadratic function f(x) = ax2 + bx + c, the range depends on the direction the parabola opens and the y-coordinate of its vertex (k).

• If a > 0 (parabola opens upwards), the vertex (h, k) is the minimum point. The range is f(x) ≥ k, or in interval notation, [k, ∞).
• If a < 0 (parabola opens downwards), the vertex (h, k) is the maximum point. The range is f(x) ≤ k, or in interval notation, (-∞, k].

To find the range, first find the vertex (h, k). The y-coordinate k is the key value.

How to Write a Quadratic Function in Standard Form?

The standard form of a quadratic function is f(x) = ax2 + bx + c.

If you have a quadratic function in a different form (e.g., vertex form f(x) = a(x - h)2 + k, or factored form f(x) = a(x - r1)(x - r2)), you can convert it to standard form by expanding and simplifying:

• From Vertex Form f(x) = a(x - h)2 + k:
  1. Expand (x - h)2 to get x2 - 2hx + h2.
  2. Distribute the a: a(x2 - 2hx + h2) = ax2 - 2ahx + ah2.
  3. Add k: f(x) = ax2 - 2ahx + ah2 + k.
  4. Combine constants: Here, b = -2ah and c = ah2 + k.
• From Factored Form f(x) = a(x - r1)(x - r2):
  1. Expand (x - r1)(x - r2) using FOIL (First, Outer, Inner, Last) to get x2 - r2x - r1x + r1r2, which simplifies to x2 - (r1 + r2)x + r1r2.
  2. Distribute the a: f(x) = a(x2 - (r1 + r2)x + r1r2) = ax2 - a(r1 + r2)x + ar1r2.

How to Find a Quadratic Equation from a Graph?

To find the equation of a quadratic function from its graph, you need to identify key features:

1. Identify the Vertex (h, k): If the vertex is clearly visible, you can use the vertex form f(x) = a(x - h)2 + k.
2. Identify another point (x, y) on the parabola: Substitute the coordinates of the vertex (h, k) and the other point (x, y) into the vertex form and solve for a.
3. Write the equation: Once you have a, h, and k, write the equation in vertex form. You can then expand it to standard form if needed.

Alternatively, if the x-intercepts (roots) r1 and r2 are clear:

1. Use the factored form: f(x) = a(x - r1)(x - r2).
2. Identify another point (x, y) on the parabola (often the y-intercept or vertex).
3. Substitute the roots and the other point into the factored form and solve for a.
4. Write the equation in factored form. Expand to standard form if needed.

If you have three distinct points, you can set up a system of three linear equations using the standard form y = ax2 + bx + c and solve for a, b, and c, but this is more algebraically intensive.

How to Solve Quadratic Inequalities? What is the Solution Set?

A quadratic inequality involves expressions like ax2 + bx + c > 0, ax2 + bx + c < 0, ax2 + bx + c ≥ 0, or ax2 + bx + c ≤ 0.

Steps to solve:

1. Rearrange: Get the inequality into a form where one side is 0 (e.g., ax2 + bx + c > 0).
2. Find Critical Points (Roots): Solve the corresponding quadratic equation ax2 + bx + c = 0 to find its roots (the x-intercepts of the parabola). These roots divide the number line into intervals.
3. Test Intervals:
   • Choose a test value from within each interval created by the roots.
   • Substitute this test value into the original inequality (or the rearranged quadratic expression ax2 + bx + c).
   • Determine if the inequality is true or false for that interval.
   Alternatively, sketch the graph of the parabola. The intervals where the parabola is above the x-axis satisfy f(x) > 0, and where it's below satisfy f(x) < 0.
4. Write the Solution Set: Express the solution as an interval or a union of intervals, paying attention to whether the inequality is strict (<, >) or inclusive (≤, ≥). For inclusive inequalities, the roots themselves are part of the solution.

"What is the solution set of the quadratic inequality" refers to this interval or union of intervals.

Who Invented/Made the Quadratic Formula?

The quadratic formula was not invented by a single person but evolved over centuries through the contributions of mathematicians from various cultures.

• Ancient Babylonians (around 2000 BC): Developed methods to solve quadratic-like problems, often geometrically.
• Ancient Greeks (e.g., Euclid, Diophantus): Also explored quadratic problems geometrically and algebraically.
• Indian Mathematicians (e.g., Brahmagupta, around 7th century AD): Provided explicit rules for solving quadratic equations, including negative solutions.
• Persian Mathematician (Muhammad ibn Musa al-Khwarizmi, around 9th century AD): His work "Al-Jabr" (from which "algebra" is derived) provided systematic methods for solving quadratic equations, often by completing the square. He did not use the modern symbolic formula but described the process.
• European Mathematicians (Renaissance onwards, e.g., Viète, Descartes, Newton): Further developed algebraic notation and generalized the methods, leading to the compact symbolic form of the quadratic formula we use today. Simon Stevin (16th century) is often credited with an early version of the modern formula.

So, it's a cumulative development rather than a single invention.