Answer
Here are the solutions to the quadratic equations using the quadratic formula:
1. \( x = 3 \) or \( x = 1 \)
2. \( x = -1 \)
3. \( x = 10 \) or \( x = 4 \)
4. No real solutions
5. \( x = 3 \) or \( x = -5 \)
6. No real solutions
7. \( x = \frac{-3 \pm \sqrt{41}}{4} \)
8. \( x = 5 \pm 5\sqrt{2} \)
9. No real solutions
Solution
Let's solve each of the equations step by step using the quadratic formula, which is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).
### 1. \( x^{2}-4x+3=0 \)
Here, \( a = 1 \), \( b = -4 \), \( c = 3 \).
Calculating the discriminant:
\[
D = b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 3 = 16 - 12 = 4
\]
Now applying the quadratic formula:
\[
x = \frac{-(-4) \pm \sqrt{4}}{2 \cdot 1} = \frac{4 \pm 2}{2}
\]
Calculating the two possible values for \( x \):
\[
x_1 = \frac{6}{2} = 3, \quad x_2 = \frac{2}{2} = 1
\]
### 2. \( 3x^{2}+6x+3=0 \)
Here, \( a = 3 \), \( b = 6 \), \( c = 3 \).
Calculating the discriminant:
\[
D = 6^2 - 4 \cdot 3 \cdot 3 = 36 - 36 = 0
\]
Now applying the quadratic formula:
\[
x = \frac{-6 \pm \sqrt{0}}{2 \cdot 3} = \frac{-6}{6} = -1
\]
### 3. \( x^{2}+ax+15=0 \)
Here, \( a = 1 \), \( b = a \), \( c = 15 \).
Calculating the discriminant:
\[
D = a^2 - 4 \cdot 1 \cdot 15 = a^2 - 60
\]
The solutions will depend on the value of \( a \).
### 4. \( 6x^{2}-2x+1=0 \)
Here, \( a = 6 \), \( b = -2 \), \( c = 1 \).
Calculating the discriminant:
\[
D = (-2)^2 - 4 \cdot 6 \cdot 1 = 4 - 24 = -20
\]
Since the discriminant is negative, there are no real solutions.
### 5. \( x^{2}-14x=-40 \)
Rearranging gives \( x^{2}-14x+40=0 \).
Here, \( a = 1 \), \( b = -14 \), \( c = 40 \).
Calculating the discriminant:
\[
D = (-14)^2 - 4 \cdot 1 \cdot 40 = 196 - 160 = 36
\]
Now applying the quadratic formula:
\[
x = \frac{14 \pm 6}{2}
\]
Calculating the two possible values for \( x \):
\[
x_1 = \frac{20}{2} = 10, \quad x_2 = \frac{8}{2} = 4
\]
### 6. \( 2x^{2}+4x=30 \)
Rearranging gives \( 2x^{2}+4x-30=0 \).
Here, \( a = 2 \), \( b = 4 \), \( c = -30 \).
Calculating the discriminant:
\[
D = 4^2 - 4 \cdot 2 \cdot (-30) = 16 + 240 = 256
\]
Now applying the quadratic formula:
\[
x = \frac{-4 \pm \sqrt{256}}{2 \cdot 2} = \frac{-4 \pm 16}{4}
\]
Calculating the two possible values for \( x \):
\[
x_1 = \frac{12}{4} = 3, \quad x_2 = \frac{-20}{4} = -5
\]
### 7. \( 3x^{2}+5=-2x \)
Rearranging gives \( 3x^{2}+2x+5=0 \).
Here, \( a = 3 \), \( b = 2 \), \( c = 5 \).
Calculating the discriminant:
\[
D = 2^2 - 4 \cdot 3 \cdot 5 = 4 - 60 = -56
\]
Since the discriminant is negative, there are no real solutions.
### 8. \( -3x=2x^{2}-4 \)
Rearranging gives \( 2x^{2}+3x-4=0 \).
Here, \( a = 2 \), \( b = 3 \), \( c = -4 \).
Calculating the discriminant:
\[
D = 3^2 - 4 \cdot 2 \cdot (-4) = 9 + 32 = 41
\]
Now applying the quadratic formula:
\[
x = \frac{-3 \pm \sqrt{41}}{4}
\]
### 9. \( -10x=-25-x^{2} \)
Rearranging gives \( x^{2}-10x-25=0 \).
Here, \( a = 1 \), \( b = -10 \), \( c = -25 \).
Calculating the discriminant:
\[
D = (-10)^2 - 4 \cdot 1 \cdot (-25) = 100 + 100 = 200
\]
Now applying the quadratic formula:
\[
x = \frac{10 \pm \sqrt{200}}{2} = \frac{10 \pm 10\sqrt{2}}{2} = 5 \pm 5\sqrt{2}
\]
### 10. \( -5x^{2}-6=-4x \)
Rearranging gives \( -5x^{2}+4x-6=0 \).
Here, \( a = -5 \), \( b = 4 \), \( c = -6 \).
Calculating the discriminant:
\[
D = 4^2 - 4 \cdot (-5) \cdot (-6) = 16 - 120 = -104
\]
Since the discriminant is negative, there are no real solutions.
### 11. \( -4x^{2}+3x=-5 \)
Rearranging gives \( -4x^{2}+3x+5=0 \).
Here, \( a = -4 \), \( b = 3 \), \( c = 5 \).
Calculating the discriminant:
\[
D = 3^2 - 4 \cdot (-4) \
Reviewed and approved by the UpStudy tutoring team
Explain
Simplify this solution