Question
\begin{tabular}{|l|}\hline Example \\ \hline Express \( 2 x^{2}-12 x+3 \) in the form \( p(x-q)^{2}+r \), where \( p, q \) and \( r \) are constants to De found. \\ \hline\end{tabular}
Ask by Gibbs Norris. in Malaysia
Mar 15,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( 2x^2 - 12x + 3 = 2(x - 3)^2 - 15 \)
Solution
1. Factor out the coefficient of \( x^2 \):
\[
2x^2-12x+3 = 2\left(x^2-6x\right)+3
\]
2. Complete the square for \( x^2-6x \). Compute half the coefficient of \( x \):
\[
\left(\frac{-6}{2}\right)^2 = 9
\]
Add and subtract \( 9 \) within the parentheses:
\[
2\left(x^2-6x+9-9\right)+3 = 2\left((x^2-6x+9)-9\right)+3
\]
3. Rewrite \( x^2-6x+9 \) as \( (x-3)^2 \):
\[
2\left((x-3)^2-9\right)+3
\]
4. Distribute \( 2 \) and simplify:
\[
2(x-3)^2-18+3 = 2(x-3)^2-15
\]
Thus, the quadratic expression in the form \( p(x-q)^2+r \) is given by:
\[
2x^2-12x+3=2(x-3)^2-15
\]
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Simplify this solution Bonus Knowledge
To express \( 2x^2 - 12x + 3 \) in the form \( p(x-q)^2 + r \), we first need to complete the square. 1. Factor out the coefficient of \( x^2 \) from the first two terms: \( 2(x^2 - 6x) + 3 \). 2. Complete the square inside the parentheses. The expression \( x^2 - 6x \) can be rewritten as: \( (x - 3)^2 - 9 \) since \( 9 \) is half of \( -6 \) squared. 3. Substitute back into the equation: \( 2[(x - 3)^2 - 9] + 3 \). 4. Distribute the 2: \( 2(x - 3)^2 - 18 + 3 \). 5. Combine constants: \( 2(x - 3)^2 - 15 \). Thus, we have \( p = 2 \), \( q = 3 \), and \( r = -15 \). The final expression is: \[ 2(x - 3)^2 - 15. \]
