Isolate, classify and solve
For ax² + b = c with a ≠ 0, first find q = (c − b)/a. Solve x² = q over the real numbers, then apply any stated restriction
- Undo the constant and coefficient Subtract b from both sides, then divide by a. A negative coefficient is handled by the same division \(\displaystyle \begin{aligned}ax^2+b&=c\\x^2&=\frac{c-b}{a}\end{aligned}\)
- Count distinct real roots q > 0 gives two opposite roots; q = 0 gives the single root zero; q < 0 gives no real roots
- Exact roots For q ≥ 0, x = ±√q. Simplify square factors. In x = ±k√n, use k > 0 and squarefree integer n > 1 when the root is irrational
- Approximate roots Round only when requested. Keep fractions exact through the calculation, then round each final root to the stated decimal places
- Restrictions A restriction filters the equation’s roots. x > 0 keeps positive roots, x < 0 keeps negative roots, and strict inequalities exclude zero
Worked example
Solve exactly over the real numbers, then apply x < 0.
\[3x^2+7=67,\quad x<0\]
- \(\displaystyle 3x^2=60\)Subtract 7 from both sides
- \(\displaystyle x^2=20\)Divide by 3.
- \(\displaystyle \begin{aligned}x&=\pm\sqrt{20}\\&=\pm2\sqrt5\end{aligned}\)20 = 4 × 5. Take both branches and simplify the square factor
- \(\displaystyle x=-2\sqrt5\)Only the negative branch satisfies x < 0.
- \(\displaystyle \begin{aligned}3(-2\sqrt5)^2+7&=3(20)+7\\&=67\end{aligned}\)Substitution verifies the original equation and the sign restriction
Common mistakes
Forgetting the other root
✕\(\displaystyle x^2=9\Rightarrow x=3\text{ only}\)✓\(\displaystyle x=-3\text{ or }x=3\)
Both signed values square to 9.Taking roots too early
✕\(\displaystyle 3x^2+7=55\Rightarrow x=\sqrt{55}-7\)✓\(\displaystyle \begin{aligned}3x^2&=48\\x^2&=16\\x&=\pm4\end{aligned}\)
Subtract the constant and divide by the coefficient before taking the root of the isolated valueInventing real roots
✕\(\displaystyle x^2=-4\Rightarrow x=\pm2\)✓\(\displaystyle \text{No real solution}\)
Both 2 and −2 square to positive 4.Counting zero twice
✕\(\displaystyle \begin{gathered}x=+0\text{ or }x=-0\\\text{two roots}\end{gathered}\)✓\(\displaystyle x=0\quad\text{one root}\)
The two signs name the same numberDifficulty levels
introTwo integer roots in separate negative and positive fields
standardSimplest exact surd data or both roots rounded to one decimal place
challengeIsolate a collected square equation and apply a positive or negative restriction
examClassify two, one or no distinct real roots and select the complete solution statement
Questions about this topic
Does √9 mean ±3?+
No. √9 is the principal, non-negative square root 3. The equation x² = 9 has two solutions, −3 and 3.
Why does a negative isolated value give no real roots?+
Every real number squared is non-negative. No real square can equal a negative number
How do I fill the exact surd table?+
Simplify the radical first, use a positive coefficient and a squarefree integer radicand. An unwritten coefficient is 1. For ±√10/3 enter coefficient 1/3 and radicand 10.
Should I round intermediate fractions?+
No. Keep them exact until the final square root has been evaluated, then round once to the requested precision
Can a restriction remove every solution?+
Yes. For example, x² = 0 has root zero, but no root satisfies x > 0.
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 10: Quadratic and non-linear graphs
In this chapter
Expand binomial products and special products10A Expanding and special products: Verify binomial and special-product expansions10B Common factors and difference of squares: Factorise common factors and differences of squares10B Common factors and difference of squares: Verify common-factor and square-difference products10C Quadratic factorisation and fractions: Factorise a quadratic expression completelySimplify an algebraic fraction by factorising and cancelling10C Quadratic factorisation and fractions: Verify monic and simple non-monic factorisationsVerify quadratic fraction cancellations and exact valuesSolve a quadratic equation by taking square rootsSolve a quadratic equation using the null factor lawSet up and solve a quadratic application problemIdentify and interpret parabola features10G Parabola features and transformations: Distinguish linear and quadratic relationsRead and use circle equations and graphsRead and use exponential growth and decay graphs10H Circles and exponential graphs: Classify a relation as linear, quadratic, circle or exponentialConnect reciprocal rules, tables and graphs