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The solutions are \( x = -2.5 \) and \( x = 2.5 \).
Solution
Solve the equation \( 3x^{2}-21=4-x^{2} \).
Solve the quadratic equation by following steps:
- step0: Solve using square roots:
\(3x^{2}-21=4-x^{2}\)
- step1: Add or subtract both sides:
\(4x^{2}=25\)
- step2: Divide both sides:
\(\frac{4x^{2}}{4}=\frac{25}{4}\)
- step3: Divide the numbers:
\(x^{2}=\frac{25}{4}\)
- step4: Simplify the expression:
\(x=\pm \sqrt{\frac{25}{4}}\)
- step5: Simplify the expression:
\(x=\pm \frac{5}{2}\)
- step6: Separate into possible cases:
\(\begin{align}&x=\frac{5}{2}\\&x=-\frac{5}{2}\end{align}\)
- step7: Rewrite:
\(x_{1}=-\frac{5}{2},x_{2}=\frac{5}{2}\)
The solutions to the equation \(3x^{2}-21=4-x^{2}\) are \(x=-\frac{5}{2}\) and \(x=\frac{5}{2}\), which can also be written as \(x=-2.5\) and \(x=2.5\).
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Simplify this solution Bonus Knowledge
To solve the equation \( 3 x^{2}-21=4-x^{2} \), we start by moving all terms to one side to set the equation to zero: \[ 3x^2 + x^2 - 21 - 4 = 0 \] This simplifies to: \[ 4x^2 - 25 = 0 \] Now, we can add 25 to both sides: \[ 4x^2 = 25 \] Next, divide both sides by 4: \[ x^2 = \frac{25}{4} \] Taking the square root of both sides gives us: \[ x = \pm \frac{5}{2} \] Thus, the solutions are: \[ x = \frac{5}{2} \quad \text{and} \quad x = -\frac{5}{2} \] The solutions can also be expressed as decimal values: \( 2.5 \) and \( -2.5 \). Happy solving!
