Question

solve $4x^2 + 25 = 75$ using square roots. round your solutions to the nearest hundredth.
$\square\\ x \approx -14.14$ $\square\\ x \approx -2.12$ $\square\\ x \approx 0.87$ $\square\\ x \approx 4.78$
$\square\\ x \approx -5.00$ $\square\\ x \approx -1.48$ $\square\\ x \approx 1.48$ $\square\\ x \approx 5.00$
$\square\\ x \approx -4.78$ $\square\\ x \approx -0.87$ $\square\\ x \approx 2.12$ $\square\\ x \approx 14.14$
$\square\\ x \approx -3.54$ $\square\\ x \approx 0$ $\square\\ x \approx 3.54$ $\square$ no solutions

Explanation:

Step1: Isolate the \(x^2\) term

Subtract 25 from both sides of the equation \(4x^2 + 25 = 75\).

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Step2: Solve for \(x^2\)

Divide both sides by 4 to get \(x^2\) by itself.

$$ x^2=\frac{50}{4} = 12.5 $$

Step3: Take square roots

Take the square root of both sides. Remember that when taking the square root to solve an equation, we consider both the positive and negative roots.

$$ x=\pm\sqrt{12.5} $$

Step4: Calculate the numerical values

Calculate the value of \(\sqrt{12.5}\). \(\sqrt{12.5}\approx3.54\) (rounded to the nearest hundredth). So the solutions are \(x\approx3.54\) and \(x\approx - 3.54\).

Answer:

\(x\approx - 3.54\), \(x\approx3.54\)