QuestionSolve for $x$ by completing the square in the equation: $7 x^{2}=4 x+5$ .

Studdy Solution

STEP 1

Assumptions1. We are given the equation $7x^{}=4x+5$ . . We are asked to solve for $x$ by completing the square.

STEP 2

First, we need to rewrite the equation in the form of $ax^{2} + bx + c =0$ . This is the standard form of a quadratic equation.
$7x^{2} -4x -5 =0$

STEP 3

Next, we divide the entire equation by $a$ (which is7 in this case) to make the coefficient of $x^{2}$ equal to1. This is an important step in completing the square.
$x^{2} - \frac{}{7}x - \frac{5}{7} =0$

STEP 4

Now, we need to rewrite the equation in the form of $(x - h)^{2} = k$ . To do this, we take half of the coefficient of $x$ , square it, and add it to both sides of the equation. The coefficient of $x$ is $-\frac{4}{7}$ , so half of this is $-\frac{2}{7}$ , and squaring this gives us $\frac{4}{49}$ .
$x^{2} - \frac{4}{7}x + \frac{4}{49} = \frac{}{7} + \frac{4}{49}$

STEP 5

The left side of the equation can now be written as a perfect square, and the right side can be simplified.
$(x - \frac{2}{7})^{2} = \frac{35}{49} + \frac{4}{49}$

STEP 6

implify the right side of the equation.
$(x - \frac{2}{})^{2} = \frac{39}{49}$

STEP 7

Now, we take the square root of both sides of the equation to solve for $x$ . Remember that when we take the square root of both sides, we must consider both the positive and negative roots.
$x - \frac{2}{7} = \pm \sqrt{\frac{39}{49}}$

STEP 8

implify the right side of the equation.
$x - \frac{2}{7} = \pm \frac{\sqrt{39}}{7}$

STEP 9

Finally, add $\frac{2}{7}$ to both sides of the equation to solve for $x$ .
$x = \frac{2}{7} \pm \frac{\sqrt{39}}{7}$

STEP 10

implify the right side of the equation.
$x = \frac{2 \pm \sqrt{39}}{7}$ So, the solutions to the equation are $x = \frac{2 + \sqrt{39}}{7}$ and $x = \frac{2 - \sqrt{39}}{7}$ .

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