What are the different ways to solve a quadratic?

A quadratic equation can be solved by factorising, using the quadratic formula or using complete the square.

What is the quadratic formula used for?

The quadratic formula calculates the roots or solutions to a quadratic equation. The formula can be used as an alternative to factorising.

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Chapter overview

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Maths

The quadratic formula - Higher

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Summary

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The quadratic formula

In a nutshell

A quadratic equation in the form $ax^2+bx+c=0$ can be solved using the quadratic formula. Sometimes, it is necessary to use the quadratic formula to solve a quadratic rather than factorising, as not all quadratics can be factorised easily.

The quadratic formula

The quadratic formula is

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

Where $a,b$ and $c$ are the co-efficients from the quadratic equation in the form

$ax^2+bx+c=0$

Substitute the values of $a, b$ and $c$ from the quadratic equation into the quadratic formula, and calculate the solutions for $x$ .

Example 1

Use the quadratic formula to solve

$2x^2+7x+3=0$

From the quadratic equation, $a=2, b=7$ and $c=3$ . Substitute the numbers into the quadratic formula.

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

$x= \dfrac {-7 \pm \sqrt {7^2-4\times2\times3}} {2 \times 2}\\$ 

Work out the value inside the square root.

$x= \dfrac {-7 \pm \sqrt {25}} {4}$ 

$x= \dfrac {-7 \pm 5} {4}\\$ 

The $\pm$ symbol means you have to do plus or minus, so there will be two calculations

$x = \dfrac {-7+5} 4 \space or \space x= \dfrac {-7-5} 4$ 

Therefore, the answers are

$\underline{x=-\frac 1 2} \space or \space \underline{x=-3}$

The quadratic formula

In a nutshell

The quadratic formula

The quadratic formula is

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

Where $a,b$ and $c$ are the co-efficients from the quadratic equation in the form

$ax^2+bx+c=0$

Substitute the values of $a, b$ and $c$ from the quadratic equation into the quadratic formula, and calculate the solutions for $x$ .

Example 1

Use the quadratic formula to solve

$2x^2+7x+3=0$

From the quadratic equation, $a=2, b=7$ and $c=3$ . Substitute the numbers into the quadratic formula.

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

$x= \dfrac {-7 \pm \sqrt {7^2-4\times2\times3}} {2 \times 2}\\$ 

Work out the value inside the square root.

$x= \dfrac {-7 \pm \sqrt {25}} {4}$ 

$x= \dfrac {-7 \pm 5} {4}\\$ 

The $\pm$ symbol means you have to do plus or minus, so there will be two calculations

$x = \dfrac {-7+5} 4 \space or \space x= \dfrac {-7-5} 4$ 

Therefore, the answers are

$\underline{x=-\frac 1 2} \space or \space \underline{x=-3}$

Want to find out more? Check out these other lessons!

Factorising quadratics

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Frequently Asked Questions (FAQ)

FAQs

Question: What are the different ways to solve a quadratic?

Answer: A quadratic equation can be solved by factorising, using the quadratic formula or using complete the square.
Question: What is the quadratic formula used for?

Answer: The quadratic formula calculates the roots or solutions to a quadratic equation. The formula can be used as an alternative to factorising.
Question: When do you use the quadratic formula to solve a quadratic?

Answer: Sometimes, it is necessary to use the quadratic formula to solve a quadratic rather than factorising, as not all quadratics can be factorised easily.

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The quadratic formula - Higher

The quadratic formula

​​In a nutshell

The quadratic formula

Example 1

The quadratic formula

​​In a nutshell

The quadratic formula

Example 1

Want to find out more? Check out these other lessons!

Now you’ve read the summary, have a go at the exercises!

In a nutshell

In a nutshell