Find the values of $p$ that will make the quadratic $(5p - 1)x^2 - 4x + (2p - 1)$ a perfect square.

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Algebra 1

Complete the square

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Q. Find the values of

p

that will make the quadratic

(5p - 1)x^2 - 4x + (2p - 1)

a perfect square.

Identify Form: Identify the general form of a perfect square quadratic. $\newline$ A perfect square quadratic is of the form $(ax + b)^2 = ax^2 + 2abx + b^2$ .
Compare with General Form: Compare the given quadratic with the general form. $\newline$ The given quadratic is 5p - 1)x^2 - 4x + (2p - 1)\. We need to find the value of \$p such that the quadratic becomes a perfect square.
Determine Coefficient of $x$ : Determine the coefficient of $x$ in the perfect square. For the quadratic to be a perfect square, the linear coefficient $(2ab)$ must be equal to $-4$ . In our case, $a = \sqrt{5p - 1}$ and $b$ would be the square root of the constant term, which we will find later.
Solve for b: Solve for b in terms of $p$ . We have $2ab = -4$ , so $2 \times \sqrt{5p - 1} \times b = -4$ . Solving for b gives us $b = \frac{-4}{2 \times \sqrt{5p - 1}}$ .
Express as $b^2$ : Express the constant term as $b^2$ . The constant term in the perfect square, $b^2$ , must be equal to $(2p - 1)$ . So, $(-4 / (2 * \sqrt{5p - 1}))^2 = 2p - 1$ .
Simplify to Solve for p: Simplify the equation to solve for p. $(-4 / (2 * \sqrt{5p - 1}))^2 = 2p - 1$ simplifies to $(16 / (4 * (5p - 1))) = 2p - 1$ . Further simplification gives us $4 / (5p - 1) = 2p - 1$ .
Clear Denominator: Multiply both sides by $(5p - 1)$ to clear the denominator. $4 = (2p - 1)(5p - 1)$ . This expands to $4 = 10p^2 - 5p - 2p + 1$ .
Combine and Solve for $p$ : Combine like terms and bring all terms to one side to solve for $p$ . $10p^2 - 7p - 3 = 0$ .
Factor Quadratic Equation: Factor the quadratic equation to find the values of $p$ . We need to find two numbers that multiply to $(10 \times -3) = -30$ and add up to $-7$ . These numbers are $-10$ and $3$ . However, the quadratic does not factor easily, so we will use the quadratic formula instead.
Apply Quadratic Formula: Apply the quadratic formula to find $p$ . $p = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 10 \cdot (-3)}}{2 \cdot 10}$ .
Simplify Quadratic Formula: Simplify the quadratic formula. $\newline$ $p = \frac{7 \pm \sqrt{49 + 120}}{20}$ . $\newline$ $p = \frac{7 \pm \sqrt{169}}{20}$ . $\newline$ $p = \frac{7 \pm 13}{20}$ .
Find Possible Values for p: Find the two possible values for $p$ . $p = \frac{7 + 13}{20}$ or $p = \frac{7 - 13}{20}$ . $p = \frac{20}{20}$ or $p = \frac{-6}{20}$ . $p = 1$ or $p = -\frac{3}{10}$ .