Quadratic Equations and Graphs: Algebra 1 Chapter 7

Why this chapter matters

Quadratics model area, projectile motion, and optimization while introducing equations that can have two, one, or no real solutions.

What you will learn

Choose an efficient method for solving a quadratic equation.
Use the discriminant to predict the number of real solutions.
Interpret roots, vertex, axis of symmetry, and intercepts on a graph.

Understand the core ideas

Quadratic expressions have highest power 2, and their graphs are parabolas. Solving a quadratic equation means finding x-values where the parabola reaches y = 0. Factoring is fastest when factors are clear, completing the square reveals vertex form and symmetry, and the quadratic formula works for every quadratic. Good strategy starts with rearranging to standard form $ax^2 +$ bx + c = 0, then selecting the method that minimizes arithmetic complexity.

The discriminant $b^2 - 4$ ac predicts root behavior before you solve fully. If it is positive, there are two distinct real roots. If zero, there is one repeated real root. If negative, there are no real roots, though complex roots exist in later study. On a graph, roots are x-intercepts, the axis of symmetry is x = -b/(2a), and the vertex gives the turning point. Linking these ideas helps you interpret equations as geometric objects, not just symbol manipulation.

Key terms

parabola: The U-shaped graph of a quadratic function.
vertex: The turning point of a parabola, maximum or minimum depending on opening.
discriminant: The value $b^2 - 4$ ac that indicates the number of real quadratic roots.
zero-product property: If ab = 0, then at least one factor must equal 0.

Solve by the quadratic formula

Solve $2x^2 - 3x - 5 = 0$ .

Identify coefficients: a = 2, b = -3, c = -5.
Compute discriminant: $b^2 - 4$ ac $= (-3)^2 - 4(2)(-5) = 9 + 40 = 49$ .
Apply formula: x = [-b +/- $\sqrt{49}$ ] / (2a) = [3 +/- 7] / 4.
Evaluate both values: x = (3 + 7)/4 = 10/4 = 5/2, and x = (3 - 7)/4 = -4/4 = -1.

Result: The solutions are x = 5/2 and x = -1.

A common misconception

Claim: Every quadratic has two different real solutions.

Correction: Some quadratics have one repeated real root or no real roots, depending on the discriminant.

Lessons in this chapter

Solving by factoringUse the zero-product property after putting the equation in standard form.
Completing the squareRewrite a quadratic in vertex form.
The quadratic formulaSolve any quadratic and interpret the discriminant. Read the full guide →
Graphing parabolasConnect algebraic forms to roots and vertex behavior.

Study task

Model the height of a thrown object with a quadratic, identify its maximum height, and find when it returns to the ground.

Chapter checkpoint

Solve

x^2 - 5x + 6 = 0

Factor as (x - 2)(x - 3) = 0, so the solutions are x = 2 and x = 3.

Quadratic Equations and Graphs