UNIT 7 • ALGEBRA

Quadratic Equation
Quadratic Formula & Discriminant

A quadratic equation $ax^2 + bx + c = 0$ is solved by the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The discriminant $b^2 - 4ac$ says whether the roots are real, equal or complex.

THE QUADRATIC FORMULAa ≠ 0

\begin{gathered} x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\ D = b^2 - 4ac \end{gathered}

1. Read a, b, cWrite the equation as

ax^2 + bx + c = 0

first; watch the signs.

2. DiscriminantWork out

D = b^2 - 4ac

. Its sign tells you what kind of roots to expect.

3. Plus and MinusDo the formula twice, once with + and once with −, to get both roots.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Solve $2x^2 - 7x + 3 = 0$ , factorise it, and find the vertex of its graph.

STEP 01 • DISCRIMINANT

Here $a = 2$ , $b = -7$ , $c = 3$ :

D = (-7)^2 - 4(2)(3) = 49 - 24 = 25

D = 25 is a positive perfect square, so there are two different rational roots.

STEP 02 • QUADRATIC FORMULA

x = \frac{7 \pm \sqrt{25}}{2 \times 2} = \frac{7 \pm 5}{4}

x = \frac{12}{4} = 3 \quad \text{or} \quad x = \frac{2}{4} = \frac{1}{2}

STEP 03 • FACTORS AND CHECK

2x^2 - 7x + 3 = (2x - 1)(x - 3)

\alpha + \beta = 3 + \tfrac{1}{2} = \tfrac{7}{2} = -\tfrac{b}{a} \ \checkmark \qquad \alpha\beta = \tfrac{3}{2} = \tfrac{c}{a} \ \checkmark

STEP 04 • VERTEX

x = -\frac{b}{2a} = \frac{7}{4}, \qquad y = 2\left(\tfrac{7}{4}\right)^2 - 7\left(\tfrac{7}{4}\right) + 3 = -\frac{25}{8}

a = 2 > 0, so the parabola opens upwards and (7/4, −25/8) is its lowest point.

FINAL CONCLUSION

x = 3 or x = 1/2

2x^2 - 7x + 3 = (2x - 1)(x - 3) = 0

The solver below opens on this equation and draws the parabola through both roots.

LABORATORY

Interactive Quadratic Solver

LABORATORY

Quadratic Engine

Solve ax² + bx + c = 0. Fractions like 1/2 are fine.

abc

Common Questions

What is the quadratic formula?

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

solves any

ax^2 + bx + c = 0

with

a \neq 0

. In India it is also called Sridharacharya's formula.

What does the discriminant tell you?

D = b^2 - 4ac

. If D > 0 there are two different real roots (rational when D is a perfect square); if D = 0 the two roots are equal; if D < 0 there are no real roots, only two complex ones.

How do you find the vertex of a parabola?

The vertex is at

x = -\frac{b}{2a}

; put that x back into the equation to get y. It is the lowest point when a > 0 and the highest when a < 0.

What are the sum and product of the roots?

For roots α and β:

\alpha + \beta = -\frac{b}{a}

and

\alpha\beta = \frac{c}{a}

. They are a quick way to check an answer.

PreviousDerivative Calculator

Unit 7•47 of 63

NextSystem of Equations

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 7 • ALGEBRA

Quadratic Equation
Quadratic Formula & Discriminant

THE QUADRATIC FORMULAa ≠ 0

\begin{gathered} x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\ D = b^2 - 4ac \end{gathered}

1. Read a, b, cWrite the equation as

ax^2 + bx + c = 0

first; watch the signs.

2. DiscriminantWork out

D = b^2 - 4ac

. Its sign tells you what kind of roots to expect.

3. Plus and MinusDo the formula twice, once with + and once with −, to get both roots.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Solve $2x^2 - 7x + 3 = 0$ , factorise it, and find the vertex of its graph.

STEP 01 • DISCRIMINANT

Here $a = 2$ , $b = -7$ , $c = 3$ :

D = (-7)^2 - 4(2)(3) = 49 - 24 = 25

D = 25 is a positive perfect square, so there are two different rational roots.

STEP 02 • QUADRATIC FORMULA

x = \frac{7 \pm \sqrt{25}}{2 \times 2} = \frac{7 \pm 5}{4}

x = \frac{12}{4} = 3 \quad \text{or} \quad x = \frac{2}{4} = \frac{1}{2}

STEP 03 • FACTORS AND CHECK

2x^2 - 7x + 3 = (2x - 1)(x - 3)

\alpha + \beta = 3 + \tfrac{1}{2} = \tfrac{7}{2} = -\tfrac{b}{a} \ \checkmark \qquad \alpha\beta = \tfrac{3}{2} = \tfrac{c}{a} \ \checkmark

STEP 04 • VERTEX

x = -\frac{b}{2a} = \frac{7}{4}, \qquad y = 2\left(\tfrac{7}{4}\right)^2 - 7\left(\tfrac{7}{4}\right) + 3 = -\frac{25}{8}

a = 2 > 0, so the parabola opens upwards and (7/4, −25/8) is its lowest point.

FINAL CONCLUSION

x = 3 or x = 1/2

2x^2 - 7x + 3 = (2x - 1)(x - 3) = 0

The solver below opens on this equation and draws the parabola through both roots.

LABORATORY

Interactive Quadratic Solver

LABORATORY

Quadratic Engine

Solve ax² + bx + c = 0. Fractions like 1/2 are fine.

abc

Common Questions

What is the quadratic formula?

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

solves any

ax^2 + bx + c = 0

with

a \neq 0

. In India it is also called Sridharacharya's formula.

What does the discriminant tell you?

D = b^2 - 4ac

. If D > 0 there are two different real roots (rational when D is a perfect square); if D = 0 the two roots are equal; if D < 0 there are no real roots, only two complex ones.

How do you find the vertex of a parabola?

The vertex is at

x = -\frac{b}{2a}

; put that x back into the equation to get y. It is the lowest point when a > 0 and the highest when a < 0.

What are the sum and product of the roots?

For roots α and β:

\alpha + \beta = -\frac{b}{a}

and

\alpha\beta = \frac{c}{a}

. They are a quick way to check an answer.

PreviousDerivative Calculator

Unit 7•47 of 63

NextSystem of Equations

Quadratic Equation Quadratic Formula & Discriminant

Step-by-Step Worked Example

x = 3 or x = 1/2

Interactive Quadratic Solver

Quadratic Engine

Common Questions

NETZ Engine

Quadratic Equation Quadratic Formula & Discriminant

Step-by-Step Worked Example

x = 3 or x = 1/2

Interactive Quadratic Solver

Quadratic Engine

Common Questions

Quadratic Equation
Quadratic Formula & Discriminant

Quadratic Equation
Quadratic Formula & Discriminant