Question

assignment
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 )
7. ( 2n^2 - 5n - 7 = 0 )
8. ( 2v^2 - 4v + 4 = 0 )

Explanation:

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

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

Step1: Identify $a,b,c$

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

Step2: Compute discriminant

$\Delta=(-4)^2-4(1)(-21)=16+84=100$

Step3: Apply quadratic formula

$a=\frac{4\pm\sqrt{100}}{2(1)}=\frac{4\pm10}{2}$

Step4: Calculate roots

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

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

Step1: Identify $a,b,c$

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

Step2: Compute discriminant

$\Delta=(-5)^2-4(3)(3)=25-36=-11$

Step3: Apply quadratic formula

$v=\frac{5\pm\sqrt{-11}}{2(3)}=\frac{5\pm i\sqrt{11}}{6}$

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

Step1: Simplify equation (divide by 2)

$k^2 + k - 12 = 0$

Step2: Identify $a,b,c$

$a=1,\ b=1,\ c=-12$

Step3: Compute discriminant

$\Delta=(1)^2-4(1)(-12)=1+48=49$

Step4: Apply quadratic formula

$k=\frac{-1\pm\sqrt{49}}{2(1)}=\frac{-1\pm7}{2}$

Step5: Calculate roots

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

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

Step1: Identify $a,b,c$

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

Step2: Compute discriminant

$\Delta=(-3)^2-4(5)(1)=9-20=-11$

Step3: Apply quadratic formula

$v=\frac{3\pm\sqrt{-11}}{2(5)}=\frac{3\pm i\sqrt{11}}{10}$

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

Step1: Identify $a,b,c$

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

Step2: Compute discriminant

$\Delta=(5)^2-4(2)(-18)=25+144=169$

Step3: Apply quadratic formula

$v=\frac{-5\pm\sqrt{169}}{2(2)}=\frac{-5\pm13}{4}$

Step4: Calculate roots

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

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

Step1: Identify $a,b,c$

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

Step2: Compute discriminant

$\Delta=(-5)^2-4(2)(2)=25-16=9$

Step3: Apply quadratic formula

$x=\frac{5\pm\sqrt{9}}{2(2)}=\frac{5\pm3}{4}$

Step4: Calculate roots

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

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7) $2n^2 - 5n - 7 = 0$

Step1: Identify $a,b,c$

$a=2,\ b=-5,\ c=-7$

Step2: Compute discriminant

$\Delta=(-5)^2-4(2)(-7)=25+56=81$

Step3: Apply quadratic formula

$n=\frac{5\pm\sqrt{81}}{2(2)}=\frac{5\pm9}{4}$

Step4: Calculate roots

$n=\frac{5+9}{4}=\frac{7}{2},\ n=\frac{5-9}{4}=-1$

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8) $2v^2 - 4v + 4 = 0$

Step1: Simplify equation (divide by 2)

$v^2 - 2v + 2 = 0$

Step2: Identify $a,b,c$

$a=1,\ b=-2,\ c=2$

Step3: Compute discriminant

$\Delta=(-2)^2-4(1)(2)=4-8=-4$

Step4: Apply quadratic formula

$v=\frac{2\pm\sqrt{-4}}{2(1)}=\frac{2\pm2i}{2}=1\pm i$

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 $x=\frac{1}{2}$
7. $n=\frac{7}{2}$ or $n=-1$
8. $v=1+i$ or $v=1-i$