Quadratic Equations — Algebra I 练习题

Grades 9–10 中等难度 10 道题

A quadratic equation is any equation that can be written in the form ax2+bx+c=0ax^2 + bx + c = 0, where a≠0a \neq 0. Quadratic equations appear throughout algebra, physics, and engineering whenever quantities change at accelerating rates — projectile motion, area optimization, and profit modeling all rely on them.

There are three main solution methods: factoring (when the trinomial factors cleanly), completing the square (useful for converting to vertex form), and the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, which always works when the discriminant b2−4ac≥0b^2 - 4ac \geq 0.

Before solving, check whether the equation is already in standard form. Combine like terms, move everything to one side, and identify aa, bb, and cc. Always verify solutions by substituting back into the original equation.

练习的技能

  • Writing equations in standard form ax2+bx+c=0ax^2 + bx + c = 0
  • Factoring trinomials and difference of squares
  • Applying the quadratic formula
  • Checking solutions by substitution
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练习题：Quadratic Equations

说明： 仔细解答每一道题。清晰地展示所有解题过程。在指定位置或按指示在另一张纸上写出最终答案。

  1. 1.Solve by factoring: x2−5x−24=0x^2 - 5x - 24 = 0.

Quadratic Equations — 练习题（续）

  1. 2.Solve by factoring: 2x2+7x−15=02x^2 + 7x - 15 = 0.
  2. 3.Solve using the quadratic formula: x2+6x+3=0x^2 + 6x + 3 = 0. Write answers in simplest radical form.

Quadratic Equations — 练习题（续）

  1. 4.Solve using the quadratic formula: 3x2−2x−4=03x^2 - 2x - 4 = 0. Write answers in simplest radical form.
  2. 5.Solve by completing the square: x2−8x+1=0x^2 - 8x + 1 = 0.

Quadratic Equations — 练习题（续）

  1. 6.Solve by completing the square: 2x2+12x−5=02x^2 + 12x - 5 = 0.
  2. 7.The product of two consecutive positive integers is 132. Find the integers.

Quadratic Equations — 练习题（续）

  1. 8.A rectangular garden has a length that is 4 feet longer than its width. The area of the garden is 96 square feet. Find the dimensions of the garden.

Quadratic Equations — 练习题（续）

  1. 9.A ball is thrown upward from a height of 6 feet with an initial velocity of 40 feet per second. The height hh (in feet) after tt seconds is given by h=−16t2+40t+6h = -16t^2 + 40t + 6. How long does it take for the ball to hit the ground? Round to the nearest hundredth of a second.

Quadratic Equations — 练习题（续）

  1. 10.Solve the equation: x4−13x2+36=0x^4 - 13x^2 + 36 = 0. (Hint: Use substitution u=x2u = x^2.)

参考答案

  1. 1.
    Factoring: (x−8)(x+3)=0(x - 8)(x + 3) = 0
    最终答案： x=8x = 8 or x=−3x = -3
  2. 2.
    Factoring: (2x−3)(x+5)=0(2x - 3)(x + 5) = 0
    最终答案： x=32x = \frac{3}{2} or x=−5x = -5
  3. 3.
    Quadratic formula: x=−6±36−122=−6±242=−6±262=−3±6x = \frac{-6 \pm \sqrt{36 - 12}}{2} = \frac{-6 \pm \sqrt{24}}{2} = \frac{-6 \pm 2\sqrt{6}}{2} = -3 \pm \sqrt{6}
    最终答案： x=−3+6x = -3 + \sqrt{6} or x=−3−6x = -3 - \sqrt{6}
  4. 4.
    Quadratic formula: x=2±4+486=2±526=2±2136=1±133x = \frac{2 \pm \sqrt{4 + 48}}{6} = \frac{2 \pm \sqrt{52}}{6} = \frac{2 \pm 2\sqrt{13}}{6} = \frac{1 \pm \sqrt{13}}{3}
    最终答案： x=1+133x = \frac{1 + \sqrt{13}}{3} or x=1−133x = \frac{1 - \sqrt{13}}{3}
  5. 5.
    Completing the square: x2−8x=−1x^2 - 8x = -1, x2−8x+16=15x^2 - 8x + 16 = 15, (x−4)2=15(x - 4)^2 = 15, x−4=±15x - 4 = \pm \sqrt{15}, x=4±15x = 4 \pm \sqrt{15}
    最终答案： x=4±15x = 4 \pm \sqrt{15}

参考答案（续）

  1. 6.
    Divide by 2: x2+6x−52=0x^2 + 6x - \frac{5}{2} = 0, x2+6x=52x^2 + 6x = \frac{5}{2}, x2+6x+9=52+9=232x^2 + 6x + 9 = \frac{5}{2} + 9 = \frac{23}{2}, (x+3)2=232(x + 3)^2 = \frac{23}{2}, x+3=±232=±462x + 3 = \pm \sqrt{\frac{23}{2}} = \pm \frac{\sqrt{46}}{2}, x=−3±462=−6±462x = -3 \pm \frac{\sqrt{46}}{2} = \frac{-6 \pm \sqrt{46}}{2}
    最终答案： x=−6±462x = \frac{-6 \pm \sqrt{46}}{2}
  2. 7.
    Let the integers be nn and n+1n+1. n(n+1)=132n(n+1) = 132, n2+n−132=0n^2 + n - 132 = 0, (n+12)(n−11)=0(n + 12)(n - 11) = 0, n=−12n = -12 or n=11n = 11 (positive, so n=11n=11). The integers are 11 and 12.
    最终答案： 11 and 12
  3. 8.
    Let width =w= w, length =w+4= w+4. w(w+4)=96w(w+4) = 96, w2+4w−96=0w^2 + 4w - 96 = 0, (w+12)(w−8)=0(w + 12)(w - 8) = 0, w=−12w = -12 or w=8w = 8 (positive, so w=8w=8). Dimensions: width = 8 ft, length = 12 ft.
    最终答案： width = 8 ft, length = 12 ft
  4. 9.
    Set h=0h = 0: −16t2+40t+6=0-16t^2 + 40t + 6 = 0, divide by −2-2: 8t2−20t−3=08t^2 - 20t - 3 = 0, quadratic formula: t=20±400+9616=20±49616=20±43116=5±314t = \frac{20 \pm \sqrt{400 + 96}}{16} = \frac{20 \pm \sqrt{496}}{16} = \frac{20 \pm 4\sqrt{31}}{16} = \frac{5 \pm \sqrt{31}}{4}, positive root: t=5+314≈5+5.56784≈2.64t = \frac{5 + \sqrt{31}}{4} \approx \frac{5 + 5.5678}{4} \approx 2.64 seconds.
    最终答案： approximately 2.64 seconds

参考答案（续）

  1. 10.
    Let u=x2u = x^2. Then u2−13u+36=0u^2 - 13u + 36 = 0, (u−4)(u−9)=0(u - 4)(u - 9) = 0, u=4u = 4 or u=9u = 9, x2=4⇒x=±2x^2 = 4 \Rightarrow x = \pm 2, x2=9⇒x=±3x^2 = 9 \Rightarrow x = \pm 3. Solutions: x=2,−2,3,−3x = 2, -2, 3, -3.
    最终答案： x=2,−2,3,−3x = 2, -2, 3, -3
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需要避免的常见错误

  • Forgetting to set the equation equal to zero before factoring
  • Sign errors when identifying bb in the quadratic formula
  • Dropping one of the two solutions from ±\pm
  • Factoring when the trinomial is not factorable over integers

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