Quadratic Equations

What this covers

• Solving by factoring and the zero product property, including trinomials where the leading coefficient is not 1
• Solving by square roots and by completing the square, including when the coefficient of x² must be divided out first
• Using the quadratic formula accurately, and reading the discriminant b² − 4ac to predict two, one, or no real solutions
• Converting between standard form, factored form, and vertex form, and finding the vertex, axis of symmetry, and intercepts
• Graphing parabolas and connecting the roots on the graph to the algebraic solutions
• Setting up quadratics from word problems — projectile height, area with a fixed perimeter, consecutive-integer products — and rejecting solutions that don't fit the context

Where learners get stuck

Dividing both sides by x to 'simplify' an equation like x² = 5x, losing the root x = 0

Dividing by a common factor is a reliable habit from linear equations, but here the factor is a variable that can equal zero. Students rarely see the missing root because the answer x = 5 looks complete.

Setting factors equal to zero when the equation isn't equal to zero, e.g. writing (x+2)(x+3) = 6 as x + 2 = 6 and x + 3 = 6

The zero product property gets remembered as a procedure about factors rather than a fact about zero. Only zero forces one of the factors to be zero, so the equation must be rearranged to = 0 first.

Sign errors in the quadratic formula when b is negative — computing −b as negative, or losing the sign inside b²

Substituting a negative number into a formula that already contains a minus sign requires bracketing, and students typically write the substitution without brackets. The result usually still produces two answers, so the mistake isn't self-flagging.

Reading the vertex of y = (x − 4)² + 1 as (−4, 1)

Everywhere else in algebra the number you see is the number you use; vertex form is one of the first places the sign is deliberately inverted, and the reason (the whole bracket is zero when x = 4) is often skipped.

What a session looks like

A session usually opens with one equation you solve out loud so the tutor can hear which method you reach for and where the arithmetic slips. From there you work through problems together — the tutor asks what the discriminant told you before you compute the roots, or why one answer to a projectile question has to be discarded. Graph and vertex questions are described verbally, so you'll practise stating coordinates, direction of opening, and axis of symmetry in words. Sessions typically end with a couple of problems you complete on your own with the tutor listening.

Helpful to know first

• Multiplying two binomials (FOIL/distribution) and simplifying the result
• Factoring out a greatest common factor and recognising a difference of squares
• Solving linear equations, including ones with fractions and negative coefficients
• Simplifying square roots such as √48 and working with negative numbers under exponents
• Plotting points and reading coordinates from a graph

Questions

When should my child use the quadratic formula instead of factoring?

Factoring is faster when the trinomial has integer factors, which is common on classwork but not on real-world problems. The formula works every time. In sessions the tutor has students spend ten seconds checking whether the discriminant is a perfect square — if it is, factoring will work; if not, go straight to the formula.

What is completing the square actually used for if the formula solves everything?

It's how the quadratic formula is derived, and it's the method for rewriting an equation in vertex form to find the maximum or minimum. It also carries forward to circle equations in geometry and to conic sections later, so it's worth doing properly rather than avoiding.

My child gets the right roots but loses marks on quadratic word problems. Why?

Usually the algebra is fine and the setup or the final step isn't: defining the variable, translating 'the product of two consecutive integers', or failing to discard a negative time or negative length. Sessions can focus specifically on setup and interpretation rather than on solving.

What does it mean when there are 'no real solutions'?

The discriminant is negative, and the parabola never crosses the x-axis. It's a genuine answer, not an error. At this level students state 'no real solutions'; in Algebra 2 the same equation gets complex solutions, so it helps to understand the graph now.