How do you complete the square for the equation x 2 + 6x = -7?

To complete the square for the equation x 2 + 6 x = -7 , follow these steps: Rewrite the equation in the form x 2 + bx = c . Add b 2 2 to both sides. Here, b is 6, so 6 2 = 3 and 3 2 = 9 . Add 9 to both sides: x 2 + 6 x + 9 = -7 + 9 . Factor the left side: x + 3 2 = 2 . Use the square root property: x + 3 = ±√2 . Solve for x : x = -3 ± √2 .

1. Equations & Inequalities

Completing the Square

1. Equations & Inequalities

Completing the Square: Videos & Practice Problems

Video Lessons Practice Worksheet

Topic summary

To solve quadratic equations like x^2 + 6x = -7, we can complete the square. This involves rewriting the equation in the form x + 62 = c. By adding 62^2 to both sides, we create a perfect square trinomial, allowing us to apply the square root property. This method is effective when the leading coefficient is 1 and the linear coefficient is even, leading to solutions like -3 ± √2.

concept

Solving Quadratic Equations by Completing the Square

Video duration:

Solving Quadratic Equations by Completing the Square Video Summary

Completing the square is a powerful technique used to solve quadratic equations that cannot be easily factored. This method transforms a quadratic equation into the form $x + a$^2 = c, allowing the use of the square root property to find solutions. To illustrate this process, consider the equation $x^2 + 6x = -7$.

First, ensure the equation is in the standard form $x^2 + bx = c$, where the leading coefficient is 1 and the constant is isolated on one side. In this case, the leading coefficient is already 1, and we can move the constant to the right side by adding 7 to both sides, resulting in $x^2 + 6x = -7$.

The next step involves adding $(\frac{b}{2})^2$ to both sides. Here, $b$ is 6, so $\frac{6}{2} = 3$ and squaring it gives us 9. Adding 9 to both sides yields:

$x^2 + 6x + 9 = -7 + 9$

which simplifies to:

$x^2 + 6x + 9 = 2$

Now, the left side can be factored as $(x + 3)^2$, leading to the equation:

$(x + 3)^2 = 2$

With the equation in the desired form, we can apply the square root property. Taking the square root of both sides gives:

$x + 3 = \pm \sqrt{2}$

To isolate $x$, subtract 3 from both sides:

$x = -3 \pm \sqrt{2}$

This results in two solutions: $x = -3 + \sqrt{2}$ and $x = -3 - \sqrt{2}$.

To further solidify understanding, let’s apply the same method to another example: $x^2 + 8x + 1 = 0$. Start by rearranging it to the form $x^2 + bx = c$ by moving the constant to the right side:

$x^2 + 8x = -1$

Next, calculate $(\frac{b}{2})^2$ where $b = 8$. Thus, $\frac{8}{2} = 4$ and squaring it gives 16. Add 16 to both sides:

$x^2 + 8x + 16 = -1 + 16$

which simplifies to:

$x^2 + 8x + 16 = 15$

Now, factor the left side as $(x + 4)^2$, resulting in:

$(x + 4)^2 = 15$

Applying the square root property, we have:

$x + 4 = \pm \sqrt{15}$

Isolating $x$ gives:

$x = -4 \pm \sqrt{15}$

Thus, the solutions are $x = -4 + \sqrt{15}$ and $x = -4 - \sqrt{15}$. Completing the square is a reliable method for solving quadratic equations, especially when factoring is not straightforward.

Problem

Solve the given quadratic equation by completing the square. $x^2+3x-5=0$

$x=-\frac32,x=\frac52$

$x=-\frac32,x=\sqrt{29}$

$x=\frac{-3+\sqrt{29}}{2},x=\frac{-3-\sqrt{29}}{2}$

$x=\frac{3+\sqrt{29}}{2},x=\frac{3-\sqrt{29}}{2}$

Problem

Solve the given quadratic equation by completing the square.
$3x^2-6x-9=0$

$x=3,x=-1$

$x=3,x=1$

$x=2,x=3$

$x=-3,x=-4$

Do you want more practice?

We have more practice problems on Completing the Square

Here’s what students ask on this topic:

What is completing the square in algebra?

Completing the square is a method used to solve quadratic equations by rewriting them in the form ${x + a}^{2} = c$ . This involves adding a specific value to both sides of the equation to create a perfect square trinomial. For example, for the equation $x^{2} + 6 x = -7$ , you add ${\frac{b}{2}}^{2}$ to both sides, where $b$ is the coefficient of $x$ . This method is particularly useful when the leading coefficient is 1 and the linear coefficient is even.

How do you complete the square for the equation x² + 6x = -7?

To complete the square for the equation $x^{2} + 6 x = -7$ , follow these steps:

Rewrite the equation in the form $x^{2} + bx = c$ .
Add ${\frac{b}{2}}^{2}$ to both sides. Here, $b$ is 6, so $\frac{6}{2} = 3$ and $3^{2} = 9$ . Add 9 to both sides: $x^{2} + 6 x + 9 = -7 + 9$ .
Factor the left side: ${x + 3}^{2} = 2$ .
Use the square root property: $x + 3 = ±√2$ .
Solve for $x$ : $x = -3 ± √2$ .

When is completing the square a good method to use?

Completing the square is a good method to use when solving quadratic equations, especially when the leading coefficient (the coefficient of $x^{2}$ ) is 1 and the linear coefficient (the coefficient of $x$ ) is even. This method is always applicable, but it is particularly efficient in these cases because it simplifies the process of creating a perfect square trinomial, which can then be solved using the square root property.

What are the steps to complete the square?

The steps to complete the square are:

Rewrite the quadratic equation in the form $x^{2} + bx = c$ .
Add ${\frac{b}{2}}^{2}$ to both sides of the equation.
Factor the left side to form a perfect square trinomial: ${x + \frac{b}{2}}^{2}$ .
Use the square root property to solve for $x$ : $x + \frac{b}{2} = ±√ c$ .
Solve for $x$ by isolating it on one side of the equation.

Why does completing the square work?

Completing the square works because it transforms a quadratic equation into a form that can be easily solved using the square root property. By adding ${\frac{b}{2}}^{2}$ to both sides, we create a perfect square trinomial, which factors into ${x + \frac{b}{2}}^{2}$ . This allows us to apply the square root property, simplifying the process of solving for $x$ . Essentially, it leverages the properties of perfect squares to make the equation more manageable.