y = ax² + bx + c graphs as a parabola. A positive a opens upward, a negative a opens downward.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Roots are where it crosses
Solving ax² + bx + c = 0 finds where the graph meets the x-axis.
Two, one, or none
A parabola can cross the axis twice, touch it once, or miss it entirely.
The vertex
The turning point sits at x = −b ÷ 2a, on the axis of symmetry.
Roots are where the graph meets the x-axis
The solutions of ax² + bx + c = 0 are exactly the x-intercepts of y = ax² + bx + c. Solving an equation and finding intercepts are the same task, which is why the graph is worth sketching before solving.
Two, one, or none
A parabola can cross the x-axis twice, touch it once, or miss it entirely. Those correspond to two real roots, one repeated root, or no real roots. The picture predicts what the algebra will produce.
The sign of a sets the direction
If a > 0 the parabola opens upwards and has a minimum; if a < 0 it opens downwards and has a maximum. Knowing which before you start tells you whether the vertex you compute should be the lowest or highest point.
Factored form shows the roots
y = (x − 2)(x − 3) has roots at 2 and 3, readable directly. This is why factoring is useful: it converts the equation into a form where the answer requires no further work.
Step 2: Try It Yourself
Change the coefficients and watch the roots appear and vanish.
- Point(2, 0)
Step 3: Watch an Example
One step at a time.
Watch Maya Find the Roots of x² − 5x + 6
Maya solves a quadratic by factoring.
- Step 1
She factors it: two numbers multiplying to 6 and adding to −5 are −2 and −3.
Step 4: Your Turn
Practice makes it stick.
The Roots
Problem 1 of 2
(x − 4)(x − 7) = 0. What is the larger root?
The Vertex
Problem 2 of 2
y = x² − 6x + 5. What is the x-coordinate of the vertex?
Roots and Turning Points
1 of 4
x² − 9 = 0. What is the positive root?
2 of 4
(x − 1)(x − 6) = 0. What is the larger root?
3 of 4
y = x² − 8x + 3. x-coordinate of the vertex?
4 of 4
x² + 5x + 6 = 0. What is the larger root?
Step 5: Quick Check
Show what you know.
Question 1 of 2
x² − 7x + 12 = 0. What is the larger root?
Question 2 of 2
What do the roots of a quadratic correspond to on its graph?
What You Learned
- A quadratic graphs as a parabola.
- Its roots are where the graph crosses the x-axis.
- The vertex sits at x = −b ÷ 2a.