Problem

Problem

Use the quadratic formula to solve exactly for xx : a. a x2−4x−3=0x^2 - 4x - 3 = 0 b. b x2+6x+7=0x^2 + 6x + 7 = 0 c. c x2+1=4xx^2 + 1 = 4x d. d x2+4x=1x^2 + 4x = 1 e. e x2−4x+2=0x^2 - 4x + 2 = 0 f. f 2x2−2x−3=02x^2 - 2x - 3 = 0 g. g 3x2−5x−1=03x^2 - 5x - 1 = 0 h. h −x2+4x+6=0-x^2 + 4x + 6 = 0 i. i −2x2+7x−2=0-2x^2 + 7x - 2 = 0
Intuitive explanation

The quadratic formula is your universal key for solving any equation of the form ax2+bx+c=0ax^2 + bx + c = 0—it works when factoring is messy or impossible. The insight is that you just need to identify your three coefficients, substitute them carefully into x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, and simplify. The ±\pm reminds you that quadratics typically have two solutions.

Intuitive explanation

The quadratic formula is your universal key for solving any equation of the form ax2+bx+c=0ax^2 + bx + c = 0—it works when factoring is messy or impossible. The insight is that you just need to identify your three coefficients, substitute them carefully into x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, and simplify. The ±\pm reminds you that quadratics typically have two solutions.

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