Quadratic Equations — Algebra I 練習ワークシート

Grades 9–10 ふつう 10問

A quadratic equation is any equation that can be written in the form $ax^2 + bx + c = 0$ , where $a \neq 0$ . 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 $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , which always works when the discriminant $b^2 - 4ac \geq 0$ .

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

練習するスキル

Writing equations in standard form $ax^2 + bx + c = 0$
Factoring trinomials and difference of squares
Applying the quadratic formula
Checking solutions by substitution

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練習ワークシート：Quadratic Equations

指示： 各問題を丁寧に解きましょう。途中式もはっきりと書きましょう。最終的な答えは、指定された場所または指示に従って別の用紙に書いてください。

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

Quadratic Equations — 練習ワークシート（続き）

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

Quadratic Equations — 練習ワークシート（続き）

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

Quadratic Equations — 練習ワークシート（続き）

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

Quadratic Equations — 練習ワークシート（続き）

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 — 練習ワークシート（続き）

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

Quadratic Equations — 練習ワークシート（続き）

10.Solve the equation: $x^4 - 13x^2 + 36 = 0$ . (Hint: Use substitution $u = x^2$ .)

解答集

1.

Factoring: $(x - 8)(x + 3) = 0$

最終回答： $x = 8$ or $x = -3$
2.

Factoring: $(2x - 3)(x + 5) = 0$

最終回答： $x = \frac{3}{2}$ or $x = -5$
3.

Quadratic formula: $x = \frac{-6 \pm \sqrt{36 - 12}}{2} = \frac{-6 \pm \sqrt{24}}{2} = \frac{-6 \pm 2\sqrt{6}}{2} = -3 \pm \sqrt{6}$

最終回答： $x = -3 + \sqrt{6}$ or $x = -3 - \sqrt{6}$
4.

Quadratic formula: $x = \frac{2 \pm \sqrt{4 + 48}}{6} = \frac{2 \pm \sqrt{52}}{6} = \frac{2 \pm 2\sqrt{13}}{6} = \frac{1 \pm \sqrt{13}}{3}$

最終回答： $x = \frac{1 + \sqrt{13}}{3}$ or $x = \frac{1 - \sqrt{13}}{3}$
5.

Completing the square: $x^2 - 8x = -1$ , $x^2 - 8x + 16 = 15$ , $(x - 4)^2 = 15$ , $x - 4 = \pm \sqrt{15}$ , $x = 4 \pm \sqrt{15}$

最終回答： $x = 4 \pm \sqrt{15}$

解答集（続き）

6.

Divide by 2: $x^2 + 6x - \frac{5}{2} = 0$ , $x^2 + 6x = \frac{5}{2}$ , $x^2 + 6x + 9 = \frac{5}{2} + 9 = \frac{23}{2}$ , $(x + 3)^2 = \frac{23}{2}$ , $x + 3 = \pm \sqrt{\frac{23}{2}} = \pm \frac{\sqrt{46}}{2}$ , $x = -3 \pm \frac{\sqrt{46}}{2} = \frac{-6 \pm \sqrt{46}}{2}$

最終回答： $x = \frac{-6 \pm \sqrt{46}}{2}$
7.

Let the integers be $n$ and $n+1$ . $n(n+1) = 132$ , $n^2 + n - 132 = 0$ , $(n + 12)(n - 11) = 0$ , $n = -12$ or $n = 11$ (positive, so $n=11$ ). The integers are 11 and 12.

最終回答： 11 and 12
8.

Let width $= w$ , length $= w+4$ . $w(w+4) = 96$ , $w^2 + 4w - 96 = 0$ , $(w + 12)(w - 8) = 0$ , $w = -12$ or $w = 8$ (positive, so $w=8$ ). Dimensions: width = 8 ft, length = 12 ft.

最終回答： width = 8 ft, length = 12 ft
9.

Set $h = 0$ : $-16t^2 + 40t + 6 = 0$ , divide by $-2$ : $8t^2 - 20t - 3 = 0$ , quadratic formula: $t = \frac{20 \pm \sqrt{400 + 96}}{16} = \frac{20 \pm \sqrt{496}}{16} = \frac{20 \pm 4\sqrt{31}}{16} = \frac{5 \pm \sqrt{31}}{4}$ , positive root: $t = \frac{5 + \sqrt{31}}{4} \approx \frac{5 + 5.5678}{4} \approx 2.64$ seconds.

最終回答： approximately 2.64 seconds

解答集（続き）

10.

Let $u = x^2$ . Then $u^2 - 13u + 36 = 0$ , $(u - 4)(u - 9) = 0$ , $u = 4$ or $u = 9$ , $x^2 = 4 \Rightarrow x = \pm 2$ , $x^2 = 9 \Rightarrow x = \pm 3$ . Solutions: $x = 2, -2, 3, -3$ .

最終回答： $x = 2, -2, 3, -3$

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避けるべきよくある間違い

Forgetting to set the equation equal to zero before factoring
Sign errors when identifying $b$ 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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