Solve for k: 44k^2 - 7k - 1 = 0
Understand the Problem
The question asks to solve for 'k' in the quadratic equation. We will use the quadratic formula to solve for the values of k.
Answer
$k = \frac{1}{4}, -\frac{1}{11}$
Answer for screen readers
$k = \frac{1}{4}, -\frac{1}{11}$
Steps to Solve
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Identify a, b, and c The quadratic equation is in the form $ax^2 + bx + c = 0$. In this case, $a = 44$, $b = -7$, and $c = -1$.
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Apply the quadratic formula The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Substituting the values of $a$, $b$, and $c$ into the formula: $$k = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(44)(-1)}}{2(44)}$$
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Simplify the expression Simplify the expression inside the square root: $$k = \frac{7 \pm \sqrt{49 + 176}}{88}$$ $$k = \frac{7 \pm \sqrt{225}}{88}$$
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Evaluate the square root Evaluate the square root: $$k = \frac{7 \pm 15}{88}$$
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Find the two possible values for k Calculate the two possible values for $k$: $$k_1 = \frac{7 + 15}{88} = \frac{22}{88} = \frac{1}{4}$$ $$k_2 = \frac{7 - 15}{88} = \frac{-8}{88} = -\frac{1}{11}$$
$k = \frac{1}{4}, -\frac{1}{11}$
More Information
The quadratic formula is a general solution that works for any quadratic equation, allowing us to find the roots even when factoring is difficult or impossible.
Tips
A common mistake is to make an error when substituting the values of $a$, $b$, and $c$ into the quadratic formula, especially when dealing with negative signs. Another mistake is to incorrectly simplify the expression inside the square root or to miscalculate the square root itself. Ensure careful arithmetic to avoid these errors.
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