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The equation 4x2+7=12x-4x^{2}+7=12x is rewritten in the form of 4(xp)2+q=0-4(x-p)^{2}+q=0. What is the value of qq ?\newline(A) 2-2\newline(B) 194\frac{19}{4}\newline(C) 374\frac{37}{4}\newline(D) 1616

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Q. The equation 4x2+7=12x-4x^{2}+7=12x is rewritten in the form of 4(xp)2+q=0-4(x-p)^{2}+q=0. What is the value of qq ?\newline(A) 2-2\newline(B) 194\frac{19}{4}\newline(C) 374\frac{37}{4}\newline(D) 1616
  1. Rewrite in Standard Form: Rewrite the given equation in standard quadratic form.\newlineThe given equation is 4x2+7=12x-4x^2 + 7 = 12x. To rewrite it in standard form, we need to move all terms to one side of the equation.\newline4x212x+7=0-4x^2 - 12x + 7 = 0
  2. Complete the Square: Complete the square to rewrite the equation in the form of 4(xp)2+q=0-4(x-p)^2 + q = 0. First, we factor out the coefficient of x2x^2, which is 4-4, from the xx terms. 4(x2+3x)+7=0-4(x^2 + 3x) + 7 = 0 Next, we find the value that completes the square for the expression x2+3xx^2 + 3x. This value is (b/2)2(b/2)^2, where bb is the coefficient of xx, which is 33 in this case. x2x^200 We add and subtract x2x^211 inside the parentheses to complete the square, remembering to multiply the subtracted term by 4-4 to keep the equation balanced. x2x^233 x2x^244
  3. Simplify Further: Simplify the equation further.\newlineNow we have 4(x+32)2+9+7=0-4(x + \frac{3}{2})^2 + 9 + 7 = 0.\newlineCombine the constant terms outside the parentheses.\newline4(x+32)2+16=0-4(x + \frac{3}{2})^2 + 16 = 0
  4. Identify Value of q: Identify the value of qq in the equation 4(xp)2+q=0-4(x-p)^2 + q = 0.\newlineFrom the simplified equation, we can see that qq is the constant term, which is 1616.

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