Question Solve the equation $\sqrt{7x+2}-2x=0$ for the value of $x$ .

Studdy Solution

STEP 1

Assumptions
1. We are given the equation $\sqrt{7x + 2} - 2x = 0$ .
2. We need to solve for the variable $x$ .

STEP 2

Isolate the square root term on one side of the equation.
$\sqrt{7x + 2} = 2x$

STEP 3

Square both sides of the equation to eliminate the square root. Remember that squaring is necessary to remove the square root, but we must be cautious of potential extraneous solutions that can arise from this operation.
$(\sqrt{7x + 2})^2 = (2x)^2$

STEP 4

Carry out the squaring operation on both sides of the equation.
$7x + 2 = 4x^2$

STEP 5

Rearrange the equation to form a quadratic equation by moving all terms to one side.
$4x^2 - 7x - 2 = 0$

STEP 6

Factor the quadratic equation, if possible. If factoring is not feasible, use the quadratic formula to find the values of $x$ .
In this case, we will attempt to factor the equation first.

STEP 7

Check if the quadratic can be factored easily. Since the factors of $-2$ that add up to $-7$ are not immediately apparent, we will use the quadratic formula instead.
The quadratic formula is given by:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Where $a$ , $b$ , and $c$ are the coefficients from the quadratic equation $ax^2 + bx + c = 0$ .

STEP 8

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $4x^2 - 7x - 2 = 0$ .
$a = 4, b = -7, c = -2$

STEP 9

Substitute the coefficients into the quadratic formula.
$x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(4)(-2)}}{2(4)}$

STEP 10

Simplify the terms inside the formula.
$x = \frac{7 \pm \sqrt{49 + 32}}{8}$

STEP 11

Further simplify the terms under the square root.
$x = \frac{7 \pm \sqrt{81}}{8}$

STEP 12

Calculate the square root.
$x = \frac{7 \pm 9}{8}$

STEP 13

Solve for the two possible values of $x$ .
$x_1 = \frac{7 + 9}{8} = \frac{16}{8} = 2$
$x_2 = \frac{7 - 9}{8} = \frac{-2}{8} = -\frac{1}{4}$

STEP 14

Check for extraneous solutions by substituting the values of $x$ back into the original equation $\sqrt{7x + 2} - 2x = 0$ .
For $x_1 = 2$ :
$\sqrt{7(2) + 2} - 2(2) = \sqrt{14 + 2} - 4 = \sqrt{16} - 4 = 4 - 4 = 0$
This solution is valid.
For $x_2 = -\frac{1}{4}$ :
$\sqrt{7\left(-\frac{1}{4}\right) + 2} - 2\left(-\frac{1}{4}\right) = \sqrt{-\frac{7}{4} + 2} + \frac{1}{2}$
Since $\sqrt{-\frac{7}{4} + 2}$ does not result in a real number (the expression inside the square root is negative), this solution is not valid in the real number system.

STEP 15

Conclude that the only valid solution to the equation $\sqrt{7x + 2} - 2x = 0$ is $x = 2$ .
The solution to the equation is $x = 2$ .

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