Sure, let's solve the quadratic equation \(a^2 - 24a = -144\) by factoring.
Step 1: Move all terms to one side of the equation, so we have:
\(a^2 - 24a + 144 = 0\)
Step 2: Factor the quadratic expression \(a^2 - 24a + 144\). We want to find two numbers that multiply to \(144\) and add up to \(-24\). Those numbers are \(-12\) and \(-12\) because \((-12) \cdot (-12) = 144\) and \((-12) + (-12) = -24\).
So, we can rewrite the expression as:
\((a - 12)(a - 12) = 0\)
Step 3: Now, we have a perfect square on the left side. We can write it as:
\((a - 12)^2 = 0\)
Step 4: Take the square root of both sides:
\(\sqrt{(a - 12)^2} = \sqrt{0}\)
This simplifies to:
\(a - 12 = 0\)
Step 5: Add \(12\) to both sides to isolate \(a\):
\(a = 12\)
Now, we have our solution: \(a = 12\).
To check if this solution is correct, we can substitute \(a = 12\) back into the original equation:
\(a^2 - 24a = -144\)
\((12)^2 - 24(12) = -144\)
\(144 - 288 = -144\)
\(-144 = -144\)
The equation holds true, which means that the solution \(a = 12\) is correct.
Answers & Comments
Answer:
Sure, let's solve the quadratic equation \(a^2 - 24a = -144\) by factoring.
Step 1: Move all terms to one side of the equation, so we have:
\(a^2 - 24a + 144 = 0\)
Step 2: Factor the quadratic expression \(a^2 - 24a + 144\). We want to find two numbers that multiply to \(144\) and add up to \(-24\). Those numbers are \(-12\) and \(-12\) because \((-12) \cdot (-12) = 144\) and \((-12) + (-12) = -24\).
So, we can rewrite the expression as:
\((a - 12)(a - 12) = 0\)
Step 3: Now, we have a perfect square on the left side. We can write it as:
\((a - 12)^2 = 0\)
Step 4: Take the square root of both sides:
\(\sqrt{(a - 12)^2} = \sqrt{0}\)
This simplifies to:
\(a - 12 = 0\)
Step 5: Add \(12\) to both sides to isolate \(a\):
\(a = 12\)
Now, we have our solution: \(a = 12\).
To check if this solution is correct, we can substitute \(a = 12\) back into the original equation:
\(a^2 - 24a = -144\)
\((12)^2 - 24(12) = -144\)
\(144 - 288 = -144\)
\(-144 = -144\)
The equation holds true, which means that the solution \(a = 12\) is correct.
Step-by-step explanation:
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