Question $2 x^{2}-8 x+3=0$

Studdy Solution

STEP 1

1. The equation $2x^2 - 8x + 3 = 0$ is a quadratic equation.
2. The solutions can be found using the quadratic formula, factoring, or completing the square.
3. The coefficients are real numbers.

STEP 2

1. Identify the coefficients of the quadratic equation.
2. Use the quadratic formula to find the solutions.
3. Simplify the solutions.

STEP 3

Identify the coefficients in the quadratic equation $2x^2 - 8x + 3 = 0$ . The standard form of a quadratic equation is $ax^2 + bx + c = 0$ , where: - $a = 2$ - $b = -8$ - $c = 3$

STEP 4

Use the quadratic formula to find the solutions. The quadratic formula is given by:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Substitute the identified coefficients $a = 2$ , $b = -8$ , and $c = 3$ into the formula.

STEP 5

Calculate the discriminant, $b^2 - 4ac$ :
$b^2 - 4ac = (-8)^2 - 4 \times 2 \times 3$ $= 64 - 24$ $= 40$

STEP 6

Substitute the values into the quadratic formula:
$x = \frac{-(-8) \pm \sqrt{40}}{2 \times 2}$ $x = \frac{8 \pm \sqrt{40}}{4}$

STEP 7

Simplify the expression under the square root and the entire expression:
The square root of 40 can be simplified:
$\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}$
Substitute back into the expression for $x$ :
$x = \frac{8 \pm 2\sqrt{10}}{4}$

STEP 8

Simplify the fraction:
$x = \frac{8}{4} \pm \frac{2\sqrt{10}}{4}$ $x = 2 \pm \frac{\sqrt{10}}{2}$
Thus, the solutions are:
$x = 2 + \frac{\sqrt{10}}{2} \quad \text{and} \quad x = 2 - \frac{\sqrt{10}}{2}$
The solutions to the quadratic equation $2x^2 - 8x + 3 = 0$ are:
$x = 2 + \frac{\sqrt{10}}{2} \quad \text{and} \quad x = 2 - \frac{\sqrt{10}}{2}$

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