Practice with Solving Quadratic Equations Topic Index | Algebra Index | Regents Exam Prep Center

 Solve each of the following quadratic equations.   Each problem has a "hint" box if you need help getting started.  When you have solved each problem, check your answer by clicking the answer box.

 1. Solve for y:  y2 - 81= 0 Hint Factor the difference of squares Answer (y+9)(y-9)=0 y+9=0; y-9=0 y= -9; y=9 y={-9,9}

 2. Solve for m:   m2 = 7m Hint Get all terms on the same side and factor a common monomial Answer m2 - 7m = 0 m(m-7)=0 m= 0   and   m-7=0 m=0   and    m=7 m = {0,7}

 3. Solve for x:  x2 - 3x - 10 = 0 Hint Factor as a quadratic trinomial Answer (x-5)(x+2)=0 x-5=0; x+2=0 x= 5; x= -2 x = {5,-2}

 4. Solve for y:  2y2 + 4 = 9y Hint Get all terms on the left side and factor as a quadratic trinomial Answer 2y² - 9y + 4 = 0 (2y-1)(y-4)=0 2y-1=0; y-4=0 2y=1; y=4 y= 1/2; y=4 y= {1/2, 4}

 5. Solve for a:  a2 = 36 Hint Get all terms on the left and factor as the difference of squares Answer a² - 36 = 0 (a+6)(a-6)=0 a+6=0; a-6=0 a= -6; a= 6 a={-6,6}

On to the harder problems!

 6. Solve for x: Hint Cross multipy first. Answer x² = 196 x² - 196 = 0 (x+14)(x-14)=0 x+14 = 0; x-14 = 0 x = -14; x = 14 x={-14,14}

 7. Solve for x: Hint Corss multiply first. Answer 2x² + 2x - 4x - 4 = x² + 4x - x - 4 x² - 5x = 0 x(x-5) = 0 x=0; x-5 = 0 x=0; x=5 x={0,5}

 8. The square of a number is decreased by 15.  This value is twice the original number.  Find the number(s). Hint Translate into math first. Answer x² - 15 = 2x x² - 2x - 15 = 0 (x - 5)(x + 3) = 0 x - 5 = 0; x + 3 = 0 x = 5; x = -3 x = {-3, 5}

 9. The product of two consecutive integers is 56.  Find the integers. Hint Consecutive integers are represented by x and x + 1 Answer The answer is -8 and -7 or +7 and +8. x(x + 1) = 56 x² + x = 56 x² + x - 56 = 0 (x + 8)(x - 7) = 0 x + 8 = 0; x - 7 = 0 x = -8; x = 7 x = {-8,7}

 10. During practice, a softball pitcher throws a ball whose height can be modeled by the equation h = -16t 2  + 24t +1, where h = height in feet and t = time in seconds.  How long does it take for the ball to reach a height of 6 feet? Hint Replace h with 6. Answer 6 = -16t² + 24t + 1 16t² - 24t + 5 = 0 (4t - 1)(4t - 5) = 0 4t - 1 = 0; 4t - 5 = 0 4t = 1; 4t = 5 t = 1/4; t = 5/4 t = {1/4, 5/4} or {0.25, 1.25} It takes 0.25 seconds (ball going up) and 1.25 seconds (ball coming down).

 11. One leg of a right triangle exceeds the other leg by four inches.  The hypotenuse is 20 inches.  Find the length of the shorter leg of the right triangle. Hint Pythagorean Theorem Answer The legs are x and x + 4 Apply Pythagorean Theorem: x² + (x+4)² = 20² x² + x² + 8x + 16 = 400 2x² + 8x - 384 = 0 x² + 4x - 192 = 0 (x + 16)(x - 12) = 0 x = -16; x = 12 reject the negative, since length is positive. ANSWER: 12 inches

 12. A baseball diamond is a square whose side is 90 feet.  Approximate the distance from home plate to second base, to the nearest foot. Hint Pythagorean Theorem Answer 90² + 90² = x² 8100 + 8100 = x² 16200 = x² x = 127.2792206 from calculator ANSWER: 127 feet

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