Retain the complete numerator and both square-root signs.
Extended · Syllabus E2.5
Notes and key methods
For \(\displaystyle ax^{2}+bx+c=0\) with \(\displaystyle a\ne 0\), the roots are \(\displaystyle \frac{- b\pm \sqrt{b^{2}- 4ac}}{2a}\). The whole numerator is divided by 2a. Substitute the signed coefficients from a zero-on-one-side equation.
A positive discriminant gives two real roots, zero gives one repeated root, and a negative value gives no real roots. Simplify an exact surd when possible. If a decimal answer is requested, keep the exact expression until the final rounding.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · The quadratic formula and exact roots
Solve \(x^{2} + 7 x - 10 = 0\). Decimal approximations are requested only where explicitly stated.
Solve the equation exactly, simplifying any surds.
Give both roots to 3 significant figures.
Find the product of the two roots.
Find the minimum value of the quadratic expression.
Show the reasoning
(a)
\(b^2-4ac=89\) is positive. Substitution in the quadratic formula gives \(\displaystyle - \frac{\sqrt{89}}{2} - \frac{7}{2}\), \(\displaystyle - \frac{7}{2} + \frac{\sqrt{89}}{2}\). Each satisfies the original equation.
Evaluate the exact roots \(\displaystyle - \frac{\sqrt{89}}{2} - \frac{7}{2}\), \(\displaystyle - \frac{7}{2} + \frac{\sqrt{89}}{2}\) before rounding each final value to three significant figures.
Answer: \(-8.22\), \(1.22\)
(c)
Use the signed coefficients in the quadratic formula or completed-square form.
\(-10\)
Answer: \(-10\).
(d)
Use the signed coefficients in the quadratic formula or completed-square form.
\(\displaystyle - \frac{89}{4}\)
Answer: \(\displaystyle - \frac{89}{4}\).
Example 2 · The quadratic formula and exact roots
Solve \(x^{2} + 8 x - 2 = 0\). Decimal approximations are requested only where explicitly stated.
Solve the equation exactly, simplifying any surds.
Give both roots to 3 significant figures.
Find the product of the two roots.
Find the minimum value of the quadratic expression.
Show the reasoning
(a)
\(b^2-4ac=72\) is positive. Substitution in the quadratic formula gives \(- 3 \sqrt{2} - 4\), \(-4 + 3 \sqrt{2}\). Each satisfies the original equation.