Sec 3 · G1 (N(T))

Quadratic equations and functions

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1. The quadratic formula

Quadratic formula, for ax² + bx + c = 0

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

e.g.

\displaystyle 2x^{2} + 3x - 4 = 0

\displaystyle x = \dfrac{-3 \pm \sqrt{3^{2} - 4(2)(-4)}}{2(2)}

\displaystyle x = \dfrac{-3 \pm \sqrt{41}}{4}

\displaystyle x = 0.85 \text{ or } x = -2.35\text{ (2 d.p.)}

📖 Worked example

Solve $x^{2} + 6x + 2 = 0$ , giving your answers correct to 2 decimal places.

Step 1 of 3

Read off a, b and c

a = 1,\ b = 6,\ c = 2

📖 Worked example

Solve $2x^{2} - 7x + 4 = 0$ , giving your answers correct to 2 decimal places.

Step 1 of 3

Read off a, b and c

a = 2,\ b = -7,\ c = 4

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2. Word problems

Five steps

\displaystyle \text{let } x \;\to\; \text{form} \;\to\; \text{solve} \;\to\; \text{reject} \;\to\; \text{answer}

e.g.

\displaystyle x(x + 4) = 96

\displaystyle x^{2} + 4x - 96 = 0

\displaystyle (x - 8)(x + 12) = 0

\displaystyle x = 8 \text{ or } x = -12

Let $x$ be one unknown, with its unit: width $= x$ cm.
Form the equation from the story, then make one side $0$ .
Solve: factorise, or use the formula.
Reject the root that can't be: a negative length, a side that would be negative.
Answer the question asked: the length $x + 4$ , not $x$ .

📖 Worked example

The length of a rectangle is $9$ cm more than its width. Its area is $65$ cm². Let the width be $x$ cm. Find the length of the rectangle, correct to 2 decimal places.

Step 1 of 8

Which relationship gives the equation?

length × width = area

Practise this →

Practise quadratic equations and functions with step-by-step help

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1. The quadratic formula

Quadratic formula, for ax² + bx + c = 0

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

e.g.

\displaystyle 2x^{2} + 3x - 4 = 0

\displaystyle x = \dfrac{-3 \pm \sqrt{3^{2} - 4(2)(-4)}}{2(2)}

\displaystyle x = \dfrac{-3 \pm \sqrt{41}}{4}

\displaystyle x = 0.85 \text{ or } x = -2.35\text{ (2 d.p.)}

📖 Worked example

Solve $x^{2} + 6x + 2 = 0$ , giving your answers correct to 2 decimal places.

Step 1 of 3

Read off a, b and c

a = 1,\ b = 6,\ c = 2

📖 Worked example

Solve $2x^{2} - 7x + 4 = 0$ , giving your answers correct to 2 decimal places.

Step 1 of 3

Read off a, b and c

a = 2,\ b = -7,\ c = 4

Practise this →

2. Word problems

Five steps

\displaystyle \text{let } x \;\to\; \text{form} \;\to\; \text{solve} \;\to\; \text{reject} \;\to\; \text{answer}

e.g.

\displaystyle x(x + 4) = 96

\displaystyle x^{2} + 4x - 96 = 0

\displaystyle (x - 8)(x + 12) = 0

\displaystyle x = 8 \text{ or } x = -12

Let $x$ be one unknown, with its unit: width $= x$ cm.
Form the equation from the story, then make one side $0$ .
Solve: factorise, or use the formula.
Reject the root that can't be: a negative length, a side that would be negative.
Answer the question asked: the length $x + 4$ , not $x$ .

📖 Worked example

The length of a rectangle is $9$ cm more than its width. Its area is $65$ cm². Let the width be $x$ cm. Find the length of the rectangle, correct to 2 decimal places.

Step 1 of 8

Which relationship gives the equation?

length × width = area

Practise this →