Question

solve each equation with the quadratic formula.

1. ( a^2 - 4a - 21 = 0 )
2. ( 3v^2 - 5v + 3 = 0 )
3. ( 2k^2 + 2k - 24 = 0 )
4. ( 5v^2 - 3v + 1 = 0 )
5. ( 2v^2 + 5v - 18 = 0 )
6. ( 2x^2 - 5x + 2 = 0 )

Explanation:

The quadratic formula for an equation $ax^2 + bx + c = 0$ is:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

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1) Solve $a^2 - 4a - 21 = 0$

Step1: Identify $a,b,c$

Here, $a=1$, $b=-4$, $c=-21$

Step2: Substitute into formula

$$a = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-21)}}{2(1)}$$

Step3: Simplify discriminant

$$\sqrt{16 + 84} = \sqrt{100} = 10$$

Step4: Calculate solutions

$$a = \frac{4 \pm 10}{2}$$
$$a = \frac{4+10}{2}=7, \quad a = \frac{4-10}{2}=-3$$

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2) Solve $3v^2 - 5v + 3 = 0$

Step1: Identify $a,b,c$

Here, $a=3$, $b=-5$, $c=3$

Step2: Substitute into formula

$$v = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(3)}}{2(3)}$$

Step3: Simplify discriminant

$$\sqrt{25 - 36} = \sqrt{-11} = i\sqrt{11}$$

Step4: Calculate solutions

$$v = \frac{5 \pm i\sqrt{11}}{6}$$

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3) Solve $2k^2 + 2k - 24 = 0$

Step1: Simplify the equation

Divide by 2: $k^2 + k - 12 = 0$; $a=1$, $b=1$, $c=-12$

Step2: Substitute into formula

$$k = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-12)}}{2(1)}$$

Step3: Simplify discriminant

$$\sqrt{1 + 48} = \sqrt{49} = 7$$

Step4: Calculate solutions

$$k = \frac{-1 \pm 7}{2}$$
$$k = \frac{-1+7}{2}=3, \quad k = \frac{-1-7}{2}=-4$$

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4) Solve $5v^2 - 3v + 1 = 0$

Step1: Identify $a,b,c$

Here, $a=5$, $b=-3$, $c=1$

Step2: Substitute into formula

$$v = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(5)(1)}}{2(5)}$$

Step3: Simplify discriminant

$$\sqrt{9 - 20} = \sqrt{-11} = i\sqrt{11}$$

Step4: Calculate solutions

$$v = \frac{3 \pm i\sqrt{11}}{10}$$

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5) Solve $2v^2 + 5v - 18 = 0$

Step1: Identify $a,b,c$

Here, $a=2$, $b=5$, $c=-18$

Step2: Substitute into formula

$$v = \frac{-(5) \pm \sqrt{(5)^2 - 4(2)(-18)}}{2(2)}$$

Step3: Simplify discriminant

$$\sqrt{25 + 144} = \sqrt{169} = 13$$

Step4: Calculate solutions

$$v = \frac{-5 \pm 13}{4}$$
$$v = \frac{-5+13}{4}=2, \quad v = \frac{-5-13}{4}=-\frac{9}{2}$$

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6) Solve $2x^2 - 5x + 2 = 0$

Step1: Identify $a,b,c$

Here, $a=2$, $b=-5$, $c=2$

Step2: Substitute into formula

$$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(2)}}{2(2)}$$

Step3: Simplify discriminant

$$\sqrt{25 - 16} = \sqrt{9} = 3$$

Step4: Calculate solutions

$$x = \frac{5 \pm 3}{4}$$
$$x = \frac{5+3}{4}=2, \quad x = \frac{5-3}{4}=\frac{1}{2}$$

Answer:

1. $a=7$ or $a=-3$
2. $v=\frac{5 + i\sqrt{11}}{6}$ or $v=\frac{5 - i\sqrt{11}}{6}$
3. $k=3$ or $k=-4$
4. $v=\frac{3 + i\sqrt{11}}{10}$ or $v=\frac{3 - i\sqrt{11}}{10}$
5. $v=2$ or $v=-\frac{9}{2}$
6. $x=2$ or $x=\frac{1}{2}$