Algebra
Quadratic Expressions and Equations
Year 10 · Year 11
- ✓By the end of this lesson students will be able to factorise quadratic expressions of the form x² + bx + c and ax² + bx + c.
- ✓By the end of this lesson students will be able to solve quadratic equations by factorising.
- ✓By the end of this lesson students will be able to solve quadratic equations using the quadratic formula, giving exact answers or answers to a specified degree of accuracy.
- ✓By the end of this lesson students will be able to complete the square for quadratic expressions of the form x² + bx + c and ax² + bx + c (Higher Tier).
- ✓By the end of this lesson students will be able to solve quadratic equations by completing the square (Higher Tier).
Key concepts
A quadratic expression is a polynomial of degree 2. Its general form is ax² + bx + c, where a, b, and c are constants, and a ≠ 0. The highest power of the variable (usually x) is 2.
A quadratic equation is an equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Solving a quadratic equation means finding the values of x that satisfy the equation, which are also known as the roots or solutions.
To factorise a quadratic expression of the form x² + bx + c, we look for two numbers that multiply to give c and add to give b. If these numbers are p and q, then x² + bx + c = (x + p)(x + q).
To factorise a quadratic expression of the form ax² + bx + c, we can use the 'ac method'. Find two numbers that multiply to give ac and add to give b. Let these numbers be p and q. Rewrite the middle term, bx, as px + qx. Then factorise the expression by grouping the terms.
A special case of factorising is the difference of two squares. An expression of the form x² - y² can always be factorised as (x - y)(x + y).
The quadratic formula can be used to find the solutions (roots) of any quadratic equation in the form ax² + bx + c = 0. It is particularly useful when factorising is difficult or impossible.
Completing the square is a method used to rewrite a quadratic expression in the form (x + p)² + q. For x² + bx + c, this form is (x + b/2)² - (b/2)² + c. For ax² + bx + c, first factor out 'a' from the x² and x terms: a[x² + (b/a)x] + c, then complete the square for the expression inside the bracket. This form is useful for finding the turning point of a quadratic graph and for solving equations.
Key facts to remember
- 1The general form of a quadratic equation is ax² + bx + c = 0, where a ≠ 0.
- 2The solutions to a quadratic equation are called roots.
- 3To factorise x² + bx + c, find two numbers that multiply to c and add to b.
- 4The quadratic formula is x = [-b ± √(b² - 4ac)] / 2a.
- 5Completing the square rewrites x² + bx + c as (x + b/2)² - (b/2)² + c.
- 6The discriminant, b² - 4ac, tells us about the nature of the roots: >0 (two real roots), =0 (one real root), <0 (no real roots).
- 7If a quadratic equation can be factorised, it's often the quickest method to solve it.
- 8Completing the square can be used to find the turning point (vertex) of a quadratic graph.
Worked examples
Example 1
Solve the quadratic equation x² + 2x - 15 = 0 by factorising.
Answer
x = -5 or x = 3
Always check your answers by substituting them back into the original equation.
Example 2
Solve the equation 3x² - 7x - 2 = 0 using the quadratic formula. Give your answers to 2 decimal places.
Answer
x = 2.59 or x = -0.26 (to 2 d.p.)
Ensure you correctly handle negative signs when substituting into the formula, especially for b² and 4ac.
Example 3
Solve the equation x² - 8x + 10 = 0 by completing the square. Give your answers in surd form (Higher Tier).
Answer
x = 4 + √6 or x = 4 - √6
When completing the square for ax² + bx + c = 0 where a ≠ 1, first divide the entire equation by 'a' to make the coefficient of x² equal to 1.
Common mistakes
- ✗Incorrectly handling negative signs in the quadratic formula, especially for -b or -4ac.
- ✗Forgetting the ± sign when taking the square root in completing the square or when solving equations like x² = k.
- ✗Not setting the equation to 0 before applying the quadratic formula or factorising.
- ✗Making arithmetic errors when calculating b² - 4ac or simplifying fractions.
- ✗Attempting to factorise when a quadratic equation has no simple integer factors, leading to wasted time.
Exam tips
- ★Always check if a quadratic equation can be factorised first, as it's usually faster than the formula.
- ★If the question asks for answers to a specific number of decimal places or significant figures, use the quadratic formula.
- ★Memorise the quadratic formula and the process for completing the square. Write them down at the start of the exam if you're worried about forgetting.
- ★Show all your working steps clearly, especially when using the quadratic formula or completing the square, as method marks are often awarded.
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