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\( \left. \begin{array} { | l | } \hline \text { Use the quadratic formula to solve the equations. } \\ \hline 37 . - 7 x ^ { 2 } - 5 x + 1 = 0 \\ \hline \end{array} \right. \)

Ask by Munoz Ross. in the United States
Mar 19,2025

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The solutions to the equation \( -7x^2 - 5x + 1 = 0 \) are approximately \( x \approx -0.877 \) and \( x \approx 0.163 \).

Solution

To solve the quadratic equation \( -7x^2 - 5x + 1 = 0 \) using the quadratic formula, we first need to identify the coefficients \( a \), \( b \), and \( c \) from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). From the equation \( -7x^2 - 5x + 1 = 0 \): - \( a = -7 \) - \( b = -5 \) - \( c = 1 \) The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Now, let's calculate the discriminant \( b^2 - 4ac \): 1. Calculate \( b^2 \): \[ b^2 = (-5)^2 = 25 \] 2. Calculate \( 4ac \): \[ 4ac = 4 \cdot (-7) \cdot 1 = -28 \] 3. Now, calculate the discriminant: \[ b^2 - 4ac = 25 - (-28) = 25 + 28 = 53 \] Now we can substitute \( a \), \( b \), and the discriminant into the quadratic formula: \[ x = \frac{-(-5) \pm \sqrt{53}}{2 \cdot (-7)} \] This simplifies to: \[ x = \frac{5 \pm \sqrt{53}}{-14} \] Now, we can express the two solutions: 1. \( x_1 = \frac{5 + \sqrt{53}}{-14} \) 2. \( x_2 = \frac{5 - \sqrt{53}}{-14} \) Let's calculate these values. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(5+\sqrt{53}\right)}{\left(-14\right)}\) - step1: Remove the parentheses: \(\frac{5+\sqrt{53}}{-14}\) - step2: Rewrite the fraction: \(-\frac{5+\sqrt{53}}{14}\) Calculate or simplify the expression \( (5 - \sqrt(53)) / -14 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(5-\sqrt{53}\right)}{\left(-14\right)}\) - step1: Remove the parentheses: \(\frac{5-\sqrt{53}}{-14}\) - step2: Rewrite the fraction: \(-\frac{5-\sqrt{53}}{14}\) - step3: Alternative Form: \(\approx 0.162865\) The solutions to the equation \( -7x^2 - 5x + 1 = 0 \) using the quadratic formula are: 1. \( x_1 = \frac{5 + \sqrt{53}}{-14} \approx -0.877 \) 2. \( x_2 = \frac{5 - \sqrt{53}}{-14} \approx 0.163 \) Thus, the final answers are: - \( x_1 \approx -0.877 \) - \( x_2 \approx 0.163 \)

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To solve the quadratic equation \(-7x^2 - 5x + 1 = 0\) using the quadratic formula, we first need to identify the coefficients \(a\), \(b\), and \(c\). Here, \(a = -7\), \(b = -5\), and \(c = 1\). The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Plugging the values into the formula gives us: 1. Calculate the discriminant: \(b^2 - 4ac = (-5)^2 - 4(-7)(1) = 25 + 28 = 53\). 2. Substitute into the formula: \(x = \frac{-(-5) \pm \sqrt{53}}{2(-7)} = \frac{5 \pm \sqrt{53}}{-14}\). Thus, the solutions are \(x = \frac{5 + \sqrt{53}}{-14}\) and \(x = \frac{5 - \sqrt{53}}{-14}\). Now, here's an interesting nugget: the quadratic formula isn't just limited to finding roots for equations; it's a powerful tool that illustrates how mathematical concepts connect. Did you know that the quadratic formula can reveal the nature of roots? If the discriminant is positive, you have two distinct roots, but if it's zero, you'll have one double root, hinting at a vertex touch. And if it's negative, well, you’re venturing into complex numbers! When tackling quadratic equations, a common pitfall is neglecting to factor out any leading coefficients if they’re not equal to 1. For instance, in the equation given, it might be tempting to apply the formula without rewriting it in standard form. Always keep an eye out for opportunities to simplify first, as it may save you time and lead to clearer solutions.

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