Rukasha Maths

GCSE maths Beta

Quadratics: formula and completing the square · Revision guide

Higher · Algebra · 1 page in the PDF

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Learning objectives

  • Solve quadratics with the quadratic formula.
  • Complete the square and identify a turning point.

Key ideas

Write the equation in standard form; keep coefficient signs.

ax2+bx+c=0,a≠0ax^2+bx+c=0,\qquad a\ne0

The formula gives both possible real roots.

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

Positive, zero or negative discriminants give two, one repeated or no real roots.

D=b2−4acD=b^2-4ac

Halve the linear coefficient, then correct the constant.

x2+bx=(x+b2)2−(b2)2x^2+bx=\left(x+\frac b2\right)^2-\left(\frac b2\right)^2

The completed-square form shows the turning point.

y=a(x−p)2+q⇒(p,q)y=a(x-p)^2+q\quad\Rightarrow\quad(p,q)

Worked examples

Example 1

Solve using the quadratic formula.

2x2+3x−2=0⇒a=2, b=3, c=−22x^2+3x-2=0\quad\Rightarrow\quad a=2,\ b=3,\ c=-2
D=32−4(2)(−2)=25D=3^2-4(2)(-2)=25
x=−3±254=−3±54x=\frac{-3\pm\sqrt{25}}4=\frac{-3\pm5}4
x=12orx=−2x=\frac12\quad\text{or}\quad x=-2

Example 2

Complete the square.

x2−8x+7=(x−4)2−16+7x^2-8x+7=(x-4)^2-16+7
(x−4)2−9(x-4)^2-9

Watch out

Divide the whole numerator by the denominator. Keep both signs when taking a square root to solve an equation.