QuestionJuly 26, 2025

Solve the equation using the quadratic formula. 2x(x-4)=-9x+1 The solution set is square (Simplify your answer, including any radicals and i as needed. Use integers"or fractions for any numbers in the expression. Use a comma to separate answers as needed.)

Solve the equation using the quadratic formula. 2x(x-4)=-9x+1 The solution set is square (Simplify your answer, including any radicals and i as needed. Use integers"or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
Solve the equation using the quadratic formula. 2x(x-4)=-9x+1 The solution set is square (Simplify your answer, including any radicals and i as needed. Use integers"or fractions for any numbers in the expression. Use a comma to separate answers as needed.)

Solution
4.7(176 votes)

Answer

\{ \frac{1}{2}, -1 \} Explanation 1. Expand and Rearrange the Equation Start with 2x(x-4) = -9x + 1. Expand to get 2x^2 - 8x = -9x + 1. Rearrange to form a standard quadratic equation: 2x^2 + x - 1 = 0. 2. Identify Coefficients Identify a = 2, b = 1, c = -1 for the quadratic equation ax^2 + bx + c = 0. 3. Apply the Quadratic Formula Use the quadratic formula: **x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}**. Substitute a, b, and c: x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2}. 4. Calculate the Discriminant Compute the discriminant: 1^2 - 4 \cdot 2 \cdot (-1) = 1 + 8 = 9. 5. Solve for x Substitute the discriminant back into the formula: x = \frac{-1 \pm \sqrt{9}}{4}. Simplify to x = \frac{-1 \pm 3}{4}. 6. Find the Roots Calculate the roots: x_1 = \frac{-1 + 3}{4} = \frac{1}{2} and x_2 = \frac{-1 - 3}{4} = -1.

Explanation

1. Expand and Rearrange the Equation<br /> Start with $2x(x-4) = -9x + 1$. Expand to get $2x^2 - 8x = -9x + 1$. Rearrange to form a standard quadratic equation: $2x^2 + x - 1 = 0$.<br /><br />2. Identify Coefficients<br /> Identify $a = 2$, $b = 1$, $c = -1$ for the quadratic equation $ax^2 + bx + c = 0$.<br /><br />3. Apply the Quadratic Formula<br /> Use the quadratic formula: **$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$**. Substitute $a$, $b$, and $c$: $x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2}$.<br /><br />4. Calculate the Discriminant<br /> Compute the discriminant: $1^2 - 4 \cdot 2 \cdot (-1) = 1 + 8 = 9$.<br /><br />5. Solve for x<br /> Substitute the discriminant back into the formula: $x = \frac{-1 \pm \sqrt{9}}{4}$. Simplify to $x = \frac{-1 \pm 3}{4}$.<br /><br />6. Find the Roots<br /> Calculate the roots: $x_1 = \frac{-1 + 3}{4} = \frac{1}{2}$ and $x_2 = \frac{-1 - 3}{4} = -1$.
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