Question Solve for the roots of $3x^2 - 60x + 741 = 0$ using the completing the square method.

Studdy Solution

STEP 1

Assumptions
1. We are given a quadratic equation in the form $ax^2 + bx + c = 0$ .
2. We need to solve for the roots of the equation $3x^2 - 60x + 741 = 0$ .
3. We will use the method of completing the square to find the roots in simplest form.

STEP 2

First, we need to ensure that the coefficient of $x^2$ is 1. If it is not, we divide the entire equation by the coefficient of $x^2$ .
$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$

STEP 3

Divide the entire equation by the coefficient of $x^2$ , which is 3.
$\frac{3x^2}{3} - \frac{60x}{3} + \frac{741}{3} = \frac{0}{3}$

STEP 4

Simplify the equation after division.
$x^2 - 20x + 247 = 0$

STEP 5

Now, we need to complete the square. To do this, we take the coefficient of $x$ , divide it by 2, and square the result to find the number that we need to add and subtract from the left side of the equation.
$\left(\frac{b}{2}\right)^2 = \left(\frac{-20}{2}\right)^2$

STEP 6

Calculate the value to complete the square.
$\left(\frac{-20}{2}\right)^2 = (-10)^2 = 100$

STEP 7

Add and subtract this value inside the equation to complete the square.
$x^2 - 20x + 100 - 100 + 247 = 0$

STEP 8

Rewrite the equation by grouping the perfect square trinomial and the constants.
$(x^2 - 20x + 100) - 100 + 247 = 0$

STEP 9

Recognize the perfect square trinomial and rewrite it as a square of a binomial.
$(x - 10)^2 - 100 + 247 = 0$

STEP 10

Combine the constants on the left side of the equation.
$(x - 10)^2 + 147 = 0$

STEP 11

Now, we need to isolate the square term by moving the constant to the other side of the equation.
$(x - 10)^2 = -147$

STEP 12

Take the square root of both sides of the equation to solve for $x$ .
$x - 10 = \pm\sqrt{-147}$

STEP 13

Since the square root of a negative number involves the imaginary unit $i$ , we rewrite the square root.
$x - 10 = \pm\sqrt{147}i$

STEP 14

Factor out the square root of 147 to simplify it.
$x - 10 = \pm\sqrt{3 \cdot 49}i$

STEP 15

Recognize that $\sqrt{49} = 7$ and simplify the square root.
$x - 10 = \pm7\sqrt{3}i$

STEP 16

Now, add 10 to both sides to solve for $x$ .
$x = 10 \pm 7\sqrt{3}i$
The roots of the equation $3x^2 - 60x + 741 = 0$ in simplest form are $x = 10 + 7\sqrt{3}i$ and $x = 10 - 7\sqrt{3}i$ .

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