Full solution

Q. Solve using the quadratic formula.

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6x^2 - 7x + 2 = 0

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Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

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x =

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x =

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Quadratic Formula: The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$ . In this case, $a = 6$ , $b = -7$ , and $c = 2$ .
Calculate Discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: $b^2 - 4ac$ . Discriminant = $(-7)^2 - 4(6)(2) = 49 - 48 = 1$ .
Apply Quadratic Formula: Since the discriminant is positive, there will be two real solutions. Now, apply the quadratic formula to find the two values of $x$ . $x = \frac{-(-7) \pm \sqrt{1}}{2 \times 6}$ .
Solve for x (Positive): Simplify the equation by calculating the numerator for both the positive and negative square root cases. $\newline$ $x = \frac{7 \pm 1}{12}$ .
Solve for x (Negative): Now, solve for x using the positive square root: $\newline$ $x = \frac{7 + 1}{12} = \frac{8}{12}$ . $\newline$ Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $4$ . $\newline$ $x = \frac{2}{3}$ .
Solve for x (Negative): Now, solve for x using the positive square root: $\newline$ $x = \frac{7 + 1}{12} = \frac{8}{12}$ . $\newline$ Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $4$ . $\newline$ $x = \frac{2}{3}$ .Next, solve for x using the negative square root: $\newline$ $x = \frac{7 - 1}{12} = \frac{6}{12}$ . $\newline$ Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $6$ . $\newline$ $x = \frac{1}{2}$ .