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SPI Mathematics · Quadratic Equations

Solving by Factorisation and the Formula

A root is a value of x that makes the equation true; you find the roots by splitting the middle term or by the quadratic formula.

All Quadratic Equations notes

Why this matters

Four questions in the bank, three EASY and one MODERATE: check which equation has a given root, factorise, or use the formula. Every option is a pair of numbers, so a sign slip in one factor lands you on a printed wrong pair.

Concept 1 of 3: A root of a quadratic equation makes it true

A root is a value of x that balances the equation. Put it in place of x: if the left side comes to 0, it is a root; if not, it is not.

Definition

  • A root (or solution) of ax2+bx+c=0ax^2 + bx + c = 0 is a number that makes ax2+bx+cax^2 + bx + c equal to 0.
  • A quadratic equation has at most two roots.
  • If a root is given and a coefficient is unknown, put the root in and solve for the coefficient.

Condition for a root

aα2+bα+c=0a\alpha^2 + b\alpha + c = 0
  • αa root of ax² + bx + c = 0

Worked example

Is x=−2x = -2 a root of x2+7x+10=0x^2 + 7x + 10 = 0?
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Quadratic Equations · Solving by Factorisation and the Formula

A quadratic equation whose one root is 3 is

Squaring a negative number wrongly

(−2)2=4(-2)^2 = 4, not −4-4. Put a negative root in brackets before squaring it.

Concept 2 of 3: Solving a quadratic equation by splitting the middle term

If a product of two brackets is 0, one of the brackets must be 0. So turn the quadratic into two brackets and set each one to 0. To find the brackets, split the middle term into two parts that let you group.

Definition

  • First make one side 0.
  • For ax2+bx+cax^2 + bx + c, find two numbers whose product is ac and whose sum is b.
  • Split bx into those two parts, group in pairs, and take out the common bracket.
  • Set each factor to 0: (x+3)=0(x + 3) = 0 gives x=−3x = -3.

Zero product rule

(x−p)(x−q)=0⇒x=p or x=q(x - p)(x - q) = 0 \Rightarrow x = p \text{ or } x = q

Worked example

Solve 3x2−10x+8=03x^2 - 10x + 8 = 0.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Quadratic Equations · Solving by Factorisation and the Formula

The roots of the quadratic equation 2x2−x−6=02x^2 - x - 6 = 0 are

Dividing both sides by x

Dividing x2=7xx^2 = 7x by x leaves only x=7x = 7 and loses the root 0. Move everything to one side and factorise: x(x−7)=0x(x - 7) = 0.

Taking the root with the bracket's sign

The factor (x+3)(x + 3) gives the root −3-3, and (x−3)(x - 3) gives +3+3. The root has the opposite sign to the number in the bracket.

Factorising before one side is 0

In x2+3x=10x^2 + 3x = 10, move the 10 first: x2+3x−10=0x^2 + 3x - 10 = 0, which is (x+5)(x−2)=0(x + 5)(x - 2) = 0.
Drill 1 more on solving a quadratic equation by splitting the middle term

Concept 3 of 3: The quadratic formula

When no two whole numbers split the middle term, the formula still works. It gives both roots at once from a, b and c, and the part under the square root decides what kind of roots they are.

Definition

  • Read a, b and c from the standard form, signs included.
  • Work out b2−4acb^2 - 4ac first, then its square root. Simplify the root: 20=25\sqrt{20} = 2\sqrt{5}.
  • Divide every term on top by 2a.

Quadratic formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Worked example

Solve x2−4x−1=0x^2 - 4x - 1 = 0.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Quadratic Equations · Solving by Factorisation and the Formula

If 5m2−2m=25m^2 - 2m = 2 then m =

Losing the sign of b

The formula starts with −b-b. When b=−4b = -4, −b=+4-b = +4. Write b with its sign before you start.

Dividing only part of the top by 2a

4±252=2±5\frac{4 \pm 2\sqrt{5}}{2} = 2 \pm \sqrt{5}: both the 4 and the 252\sqrt{5} are divided by 2, not just one of them.

Summary — formulas & gotchas at a glance

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Formulas (3)

Watch out for (6)

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