Question
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Answer
The solutions are \(x=-8\) and \(x=1\).
Solution
Solve the equation \( 3x^{2}+21x=24 \).
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
\(3x^{2}+21x=24\)
- step1: Move the expression to the left side:
\(3x^{2}+21x-24=0\)
- step2: Factor the expression:
\(3\left(x-1\right)\left(x+8\right)=0\)
- step3: Divide the terms:
\(\left(x-1\right)\left(x+8\right)=0\)
- step4: Separate into possible cases:
\(\begin{align}&x-1=0\\&x+8=0\end{align}\)
- step5: Solve the equation:
\(\begin{align}&x=1\\&x=-8\end{align}\)
- step6: Rewrite:
\(x_{1}=-8,x_{2}=1\)
The solutions to the equation \(3x^{2}+21x=24\) are \(x=-8\) and \(x=1\).
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Simplify this solution Beyond the Answer
Let's dive into the world of quadratic equations! The equation \( 3x^{2} + 21x - 24 = 0 \) can be simplified by dividing everything by 3, leading to \( x^{2} + 7x - 8 = 0 \). Factoring this gives us \( (x + 8)(x - 1) = 0 \), leading to the solutions \( x = -8 \) and \( x = 1 \). Voilà! You've just unlocked the secrets of solving quadratics! Now, why is this relevant in the real world? Quadratic equations pop up all over the place! From projectile motion in sports to optimizing business profits, understanding how to solve them can help you make informed decisions. So, the next time you throw a ball or analyze a revenue graph, remember, you're using those quadratic skills in action!
