Question Solve the quadratic equation $6 x^{2} - 13 x + 5 = 0$ and find the four possible solutions for $x$ .

Studdy Solution

STEP 1

Assumptions
1. We are given a quadratic equation in the form $ax^2 + bx + c = 0$ .
2. The coefficients for this quadratic equation are $a = 6$ , $b = -13$ , and $c = 5$ .
3. We need to solve for $x$ .

STEP 2

The quadratic formula is used to find the roots of a quadratic equation of the form $ax^2 + bx + c = 0$ .
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

STEP 3

We will apply the quadratic formula to the given equation $6x^2 - 13x + 5 = 0$ .

STEP 4

Identify the coefficients in the given equation:
$a = 6, b = -13, c = 5$

STEP 5

Calculate the discriminant $(b^2 - 4ac)$ of the quadratic equation.
$\Delta = b^2 - 4ac$

STEP 6

Substitute the values of $a$ , $b$ , and $c$ into the discriminant formula.
$\Delta = (-13)^2 - 4 \cdot 6 \cdot 5$

STEP 7

Calculate the discriminant.
$\Delta = 169 - 120$

STEP 8

Find the value of the discriminant.
$\Delta = 49$

STEP 9

Since the discriminant is positive, we have two distinct real roots. Now, we will use the quadratic formula to find the roots.

STEP 10

Substitute the values of $a$ , $b$ , and the discriminant $\Delta$ into the quadratic formula.
$x = \frac{-(-13) \pm \sqrt{49}}{2 \cdot 6}$

STEP 11

Simplify the expression inside the quadratic formula.
$x = \frac{13 \pm 7}{12}$

STEP 12

Calculate the two possible values for $x$ by considering both the positive and negative square roots.
$x_1 = \frac{13 + 7}{12}$ $x_2 = \frac{13 - 7}{12}$

STEP 13

Calculate the first root $x_1$ .
$x_1 = \frac{20}{12}$

STEP 14

Simplify the first root.
$x_1 = \frac{5}{3}$

STEP 15

Calculate the second root $x_2$ .
$x_2 = \frac{6}{12}$

STEP 16

Simplify the second root.
$x_2 = \frac{1}{2}$

STEP 17

We have found the two roots of the quadratic equation:
$x_1 = \frac{5}{3}$ $x_2 = \frac{1}{2}$
Therefore, the correct answer is $x=\frac{1}{2}$ or $x=\frac{5}{3}$ .

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