How to Find the Vertex of a Quadratic Equation

The vertex of a quadratic equation or parabola is the highest or lowest point of that equation. It lies on the plane of symmetry of the entire parabola as well; whatever lies on the left of the parabola is a complete mirror image of whatever is on the right. If you want to find the vertex of a quadratic equation, you can either use the vertex formula, or complete the square.

1st Method: Using the Vertex Formula

1

Identify the values of a, b, and c. In a quadratic equation, the x² term = a, the x term = b, and the constant term (the term without a variable) = c. Let's say you're working with the following equation: y = x² + 9x + 18. In this example, a = 1, b = 9, and c = 18.

2

Use the vertex formula for finding the x-value of the vertex. The vertex is also the equation's axis of symmetry. The formula for finding the x-value of the vertex of a quadratic equation is x = -b/2a. Plug in the relevant values to find x. Substitute the values for a and b. Show your work:

• x=-b/2a
• x=-(9)/(2)(1)
• x=-9/2

3

Plug the x-value into the original equation to get the y-value. Now that you know the x-value, just plug it in to the original formula for the y value. You can think of the formula for finding the vertex of a quadratic function as being (x, y) = [(-b/2a), f(-b/2a)]. This just means that to get the y value, you have to find the x value based on the formula and then plug it back into the equation. Here's how you do it:

• y = x² + 9x + 18
• y = (-9/2)² + 9(-9/2) +18
• y = 81/4 -81/2 + 18
• y = 81/4 -162/4 + 72/4
• y = (81 - 162 + 72)/4
• y = -9/4

4

Write down the x and y values as an ordered pair. Now that you know that x = -9/2, and y = -9/4, just write them down as an ordered pair: (-9/2, -9/4). The vertex of this quadratic equation is (-9/2, -9/4). If you were to draw this parabola on a graph, this point would be the minimum of the parabola, because the x² term is positive.

2nd Method: Completing the Square

1

Write down the equation. Completing the square is another way to find the vertex of a quadratic equation. For this method, when you get to the end, you'll be able to find your x and y coordinates right away, instead of plugging the x coordinate back in to the original equation. Let's say you're working with the following quadratic equation: x² + 4x + 1 = 0.

2

Divide each term by the coefficient of the x² term. In this case, the coefficient of the x² term is 1, so you can skip this step. Dividing each term by 1 would not change anything.

3

Move the constant term to the right side of the equation. The constant term is the term without a coefficient. In this case, it is "1." Move 1 to the other side of the equation by subtracting 1 from both sides. Here's how you do it:

• x² + 4x + 1 = 0
• x² + 4x + 1 -1 = 0 - 1
• x² + 4x = - 1

4

Complete the square on the left side of the equation. To do this, simply find(b/2)² and add the result to both sides of the equation. Plug in "4" for b, since "4x" is the b-term of this equation.

• (4/2)² = 2² = 4. Now, add 4 to both sides of the equation to get the following:
• x² + 4x + 4 = -1 + 4
• x² + 4x + 4 = 3

5

Factor the left side of the equation. Now you will see that x² + 4x + 4 is a perfect square. It can be rewritten as (x + 2)² = 3

6

Use this format to find the x and y coordinates. You can find your x coordinate by simply setting (x + 2)² equal to zero. So when (x + 2)² = 0, what would x have to be? The variable x would have to be -2 to balance out the +2, so your x coordinate is -2. Your y-coordinate is simply the constant term on the other side of the equation. So, y = 3. You can also do a shortcut and just take the opposite sign of the number in parentheses to get the x-coordinate. So the vertex of the equation x² + 4x + 1 = (-2, -3)