Rewrite the equation by completing the square. {:[x^(2)+7x+12=0],[(x+◻)^(2)=◻]:}

Identify coefficients: Identify the quadratic and linear coefficients of the equation x^2 + 7x + 12 = 0. The quadratic coefficient is 1 (the coefficient of x^2) and the linear coefficient is 7 (the coefficient of x).Find perfect square trinomial: To complete the square, we need to find a number that, when added and subtracted to the equation, forms a perfect square trinomial. This number is (7/2)^2, which is the square of half the linear coefficient. Calculate (7/2)^2. (7/2)^2 = 49/4Calculate (7/2)^2: Rewrite the equation by adding and subtracting (7/2)^2 inside the equation. x^2 + 7x + (49/4) - (49/4) + 12 = 0Rewrite the equation: Combine the constant terms (12 and -49/4) outside the square. 12 - (49/4) = 48/4 - 49/4 = -1/4Combine constant terms: Now, write the equation with the completed square and the combined constant term. (x + 7/2)^2 - 1/4 = 0Write equation with completed square: Add 1/4 to both sides to isolate the completed square on one side. (x + 7/2)^2 = 1/4

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Math Problems

Grade 8

Negative Exponents

Full solution

Q. Rewrite the equation by completing the square.

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\begin{array}{l} x^{2}+7 x+12=0 \\ (x+\square)^{2}=\square \end{array}

Identify coefficients: Identify the quadratic and linear coefficients of the equation $x^2 + 7x + 12 = 0$ . $\newline$ The quadratic coefficient is $1$ (the coefficient of $x^2$ ) and the linear coefficient is $7$ (the coefficient of $x$ ).
Find perfect square trinomial: To complete the square, we need to find a number that, when added and subtracted to the equation, forms a perfect square trinomial. This number is $(\frac{7}{2})^2$ , which is the square of half the linear coefficient. $\newline$ Calculate $(\frac{7}{2})^2$ . $\newline$ $(\frac{7}{2})^2 = \frac{49}{4}$
Calculate $(\frac{7}{2})^2$ : Rewrite the equation by adding and subtracting $(\frac{7}{2})^2$ inside the equation. $\newline$ $x^2 + 7x + (\frac{49}{4}) - (\frac{49}{4}) + 12 = 0$
Rewrite the equation: Combine the constant terms $12$ and $-\frac{49}{4}$ outside the square. $12 - \left(\frac{49}{4}\right) = \frac{48}{4} - \frac{49}{4} = -\frac{1}{4}$
Combine constant terms: Now, write the equation with the completed square and the combined constant term. $\newline$ $(x + \frac{7}{2})^2 - \frac{1}{4} = 0$
Write equation with completed square: Add $\frac{1}{4}$ to both sides to isolate the completed square on one side. $\newline$ $\left(x + \frac{7}{2}\right)^2 = \frac{1}{4}$