solve each equation with the quadratic formula.
1) $2n^{2}-5n+2=0$
2) $2v^{2}-3v-9=0$
3) $4r^{2}+2r-6=0$
4) $n^{2}-2n-3=0$
5) $k^{2}-3k-10=0$
6) $2n^{2}+8n+8=0$
7) $2x^{2}+2x-4=0$
8) $2b^{2}+6b-56=0$
9) $2p^{2}+8p-42=0$
10) $4r^{2}+r-18=0$
11) $m^{2}-6m-16=0$
12) $4r^{2}+8r-5=0$
13) $p^{2}-5p-36=0$
14) $4r^{2}+6r-28=0$
15) $2v^{2}-6v-8=0$
16) $2x^{2}+8x-24=0$
17) $m^{2}-6m-55=0$
18) $2x^{2}+5x-25=0$
19) $4x^{2}-2x-12=0$
20) $4m^{2}-7m-15=0$

Question

Accepted Answer

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

---
## 1) $2n^2 -5n +2=0$
## 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
$n = \frac{5 \pm \sqrt{9}}{4} = \frac{5\pm3}{4}$
# Answer:
$n=2$ or $n=\frac{1}{2}$

---
## 2) $2v^2 -3v -9=0$
## Step1: Identify a,b,c
$a=2, b=-3, c=-9$
## Step2: Compute discriminant
$\Delta = (-3)^2 - 4(2)(-9) = 9+72=81$
## Step3: Apply quadratic formula
$v = \frac{3 \pm \sqrt{81}}{4} = \frac{3\pm9}{4}$
# Answer:
$v=3$ or $v=-\frac{3}{2}$

---
## 3) $4r^2 +2r -6=0$
## Step1: Simplify equation
Divide by 2: $2r^2 +r -3=0$, $a=2, b=1, c=-3$
## Step2: Compute discriminant
$\Delta = 1^2 - 4(2)(-3) = 1+24=25$
## Step3: Apply quadratic formula
$r = \frac{-1 \pm \sqrt{25}}{4} = \frac{-1\pm5}{4}$
# Answer:
$r=1$ or $r=-\frac{3}{2}$

---
## 4) $n^2 -2n -3=0$
## Step1: Identify a,b,c
$a=1, b=-2, c=-3$
## Step2: Compute discriminant
$\Delta = (-2)^2 - 4(1)(-3) = 4+12=16$
## Step3: Apply quadratic formula
$n = \frac{2 \pm \sqrt{16}}{2} = \frac{2\pm4}{2}$
# Answer:
$n=3$ or $n=-1$

---
## 5) $k^2 -3k -10=0$
## Step1: Identify a,b,c
$a=1, b=-3, c=-10$
## Step2: Compute discriminant
$\Delta = (-3)^2 - 4(1)(-10) = 9+40=49$
## Step3: Apply quadratic formula
$k = \frac{3 \pm \sqrt{49}}{2} = \frac{3\pm7}{2}$
# Answer:
$k=5$ or $k=-2$

---
## 6) $2n^2 +8n +8=0$
## Step1: Simplify equation
Divide by 2: $n^2 +4n +4=0$, $a=1, b=4, c=4$
## Step2: Compute discriminant
$\Delta = 4^2 - 4(1)(4) = 16-16=0$
## Step3: Apply quadratic formula
$n = \frac{-4 \pm \sqrt{0}}{2} = -2$
# Answer:
$n=-2$ (repeated root)

---
## 7) $2x^2 +2x -4=0$
## Step1: Simplify equation
Divide by 2: $x^2 +x -2=0$, $a=1, b=1, c=-2$
## Step2: Compute discriminant
$\Delta = 1^2 - 4(1)(-2) = 1+8=9$
## Step3: Apply quadratic formula
$x = \frac{-1 \pm \sqrt{9}}{2} = \frac{-1\pm3}{2}$
# Answer:
$x=1$ or $x=-2$

---
## 8) $2b^2 +6b -56=0$
## Step1: Simplify equation
Divide by 2: $b^2 +3b -28=0$, $a=1, b=3, c=-28$
## Step2: Compute discriminant
$\Delta = 3^2 - 4(1)(-28) = 9+112=121$
## Step3: Apply quadratic formula
$b = \frac{-3 \pm \sqrt{121}}{2} = \frac{-3\pm11}{2}$
# Answer:
$b=4$ or $b=-7$

---
## 9) $2p^2 +8p -42=0$
## Step1: Simplify equation
Divide by 2: $p^2 +4p -21=0$, $a=1, b=4, c=-21$
## Step2: Compute discriminant
$\Delta = 4^2 - 4(1)(-21) = 16+84=100$
## Step3: Apply quadratic formula
$p = \frac{-4 \pm \sqrt{100}}{2} = \frac{-4\pm10}{2}$
# Answer:
$p=3$ or $p=-7$

---
## 10) $4r^2 +r -18=0$
## Step1: Identify a,b,c
$a=4, b=1, c=-18$
## Step2: Compute discriminant
$\Delta = 1^2 - 4(4)(-18) = 1+288=289$
## Step3: Apply quadratic formula
$r = \frac{-1 \pm \sqrt{289}}{8} = \frac{-1\pm17}{8}$
# Answer:
$r=2$ or $r=-\frac{9}{4}$

---
## 11) $m^2 -6m -16=0$
## Step1: Identify a,b,c
$a=1, b=-6, c=-16$
## Step2: Compute discriminant
$\Delta = (-6)^2 - 4(1)(-16) = 36+64=100$
## Step3: Apply quadratic formula
$m = \frac{6 \pm \sqrt{100}}{2} = \frac{6\pm10}{2}$
# Answer:
$m=8$ or $m=-2$

---
## 12) $4r^2 +8r -5=0$
## Step1: Identify a,b,c
$a=4, b=8, c=-5$
## Step2: Compute discriminant
$\Delta = 8^2 - 4(4)(-5) = 64+80=144$
## Step3: Apply quadratic formula
$r = \frac{-8 \pm \sqrt{144}}{8} = \frac{-8\pm12}{8}$
# Answer:
$r=\frac{1}{2}$ or $r=-\frac{5}{2}$

---
## 13) $p^2 -5p -36=0$
## Step1: Identify a,b,c
$a=1, b=-5, c=-36$
## Step2: Compute discriminant
$\Delta = (-5)^2 - 4(1)(-36) = 25+144=169$
## Step3: Apply quadratic formula
$p = \frac{5 \pm \sqrt{169}}{2} = \frac{5\pm13}{2}$
# Answer:
$p=9$ or $p=-4$

---
## 14) $4r^2 +6r -28=0$
## Step1: Simplify equation
Divide by 2: $2r^2 +3r -14=0$, $a=2, b=3, c=-14$
## Step2: Compute discriminant
$\Delta = 3^2 - 4(2)(-14) = 9+112=121$
## Step3: Apply quadratic formula
$r = \frac{-3 \pm \sqrt{121}}{4} = \frac{-3\pm11}{4}$
# Answer:
$r=2$ or $r=-\frac{7}{2}$

---
## 15) $2v^2 -6v -8=0$
## Step1: Simplify equation
Divide by 2: $v^2 -3v -4=0$, $a=1, b=-3, c=-4$
## Step2: Compute discriminant
$\Delta = (-3)^2 - 4(1)(-4) = 9+16=25$
## Step3: Apply quadratic formula
$v = \frac{3 \pm \sqrt{25}}{2} = \frac{3\pm5}{2}$
# Answer:
$v=4$ or $v=-1$

---
## 16) $2x^2 +8x -24=0$
## Step1: Simplify equation
Divide by 2: $x^2 +4x -12=0$, $a=1, b=4, c=-12$
## Step2: Compute discriminant
$\Delta = 4^2 - 4(1)(-12) = 16+48=64$
## Step3: Apply quadratic formula
$x = \frac{-4 \pm \sqrt{64}}{2} = \frac{-4\pm8}{2}$
# Answer:
$x=2$ or $x=-6$

---
## 17) $m^2 -6m -55=0$
## Step1: Identify a,b,c
$a=1, b=-6, c=-55$
## Step2: Compute discriminant
$\Delta = (-6)^2 - 4(1)(-55) = 36+220=256$
## Step3: Apply quadratic formula
$m = \frac{6 \pm \sqrt{256}}{2} = \frac{6\pm16}{2}$
# Answer:
$m=11$ or $m=-5$

---
## 18) $2x^2 +5x -25=0$
## Step1: Identify a,b,c
$a=2, b=5, c=-25$
## Step2: Compute discriminant
$\Delta = 5^2 - 4(2)(-25) = 25+200=225$
## Step3: Apply quadratic formula
$x = \frac{-5 \pm \sqrt{225}}{4} = \frac{-5\pm15}{4}$
# Answer:
$x=\frac{5}{2}$ or $x=-5$

---
## 19) $4x^2 -2x -12=0$
## Step1: Simplify equation
Divide by 2: $2x^2 -x -6=0$, $a=2, b=-1, c=-6$
## Step2: Compute discriminant
$\Delta = (-1)^2 - 4(2)(-6) = 1+48=49$
## Step3: Apply quadratic formula
$x = \frac{1 \pm \sqrt{49}}{4} = \frac{1\pm7}{4}$
# Answer:
$x=2$ or $x=-\frac{3}{2}$

---
## 20) $4m^2 -7m -15=0$
## Step1: Identify a,b,c
$a=4, b=-7, c=-15$
## Step2: Compute discriminant
$\Delta = (-7)^2 - 4(4)(-15) = 49+240=289$
## Step3: Apply quadratic formula
$m = \frac{7 \pm \sqrt{289}}{8} = \frac{7\pm17}{8}$
# Answer:
$m=3$ or $m=-\frac{5}{4}$

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QUESTION IMAGE

Question

Explanation:

1) $2n^2 -5n +2=0$

Step1: Identify a,b,c

Step2: Compute discriminant

Step3: Apply quadratic formula

Answer:

2) $2v^2 -3v -9=0$

Step1: Identify a,b,c

Step2: Compute discriminant

Step3: Apply quadratic formula