Solve quadratic equations: methods and graphs

Algebra • Solving equations and inequalities

Flashcards

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How to handle linear terms when factorising non-monic quadratics?

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Look for factor pairs of ac that sum to b, or use splitting the middle term, then group and factor.

Key concepts

What you'll likely be quizzed about

Standard form and rearrangement

A quadratic equation takes the standard form ax^2 + bx + c = 0 with a ≠ 0. Rearrangement moves all terms to one side so the equation equals zero, which enables factorising, completing the square, and direct substitution into the quadratic formula. Rearrangement prevents missed solutions and aligns with graph interpretation because roots correspond to x-values that make y = ax^2 + bx + c equal to zero.

Factorising to solve quadratics

Factorising splits ax^2 + bx + c into two linear factors (px + q)(rx + s) when such factors exist and multiply to give the original coefficients. If (px + q)(rx + s) = 0 then px + q = 0 or rx + s = 0, which yields the roots by solving each linear equation. Factorising is efficient when integer factors appear; failure to find integer factors indicates the need for completing the square or the quadratic formula.

Completing the square

Completing the square rewrites ax^2 + bx + c into a form a(x + d)^2 + e. For a = 1, the process isolates x^2 + bx, adds (b/2)^2 to complete the square, and balances the equation by subtracting the same value. For a ≠ 1, factor a from the quadratic and linear terms first, then complete the square inside the bracket. Once in a(x + d)^2 + e = 0 form, solving becomes taking a square root: (x + d)^2 = -e/a, then x + d = ±√(-e/a). Completing the square clarifies vertex coordinates and connects algebraic solutions to graph shape.

Quadratic formula and discriminant

The quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a) provides exact roots for any quadratic ax^2 + bx + c = 0. The discriminant D = b^2 - 4ac determines the nature of roots: if D > 0 there are two distinct real roots; if D = 0 there is one repeated real root; if D < 0 there are two complex roots. The discriminant indicates whether graph-based root finding will show two x-intercepts, one tangent intercept, or no real intercepts.

Graphical solutions and approximation

A graph of y = ax^2 + bx + c is a parabola. Roots correspond to x-intercepts where y = 0. Plotting the parabola or using a graphing calculator reveals approximate x-values of intercepts when exact algebraic solutions are difficult. Approximation requires sufficient plotting resolution or calculator precision. If the parabola crosses near an integer, algebraic methods can confirm exact roots; otherwise, reading to an appropriate decimal place gives useful approximate solutions.

Key notes

Important points to keep in mind

Always rearrange to ax^2 + bx + c = 0 before algebraic methods.

If factorising, set each linear factor equal to zero to find roots.

When completing the square, balance any added terms to maintain equality.

Use the quadratic formula for exact roots when factorisation is not possible.

Use the discriminant D = b^2 - 4ac to predict number and type of roots.

Graphical methods give approximations; algebraic work gives exact values when possible.

Check solutions by substituting back into the original equation.

Include ± when taking square roots after completing the square.