In thercioes $$ 1-14 $$ solve the equation using the (ewadratic lobrmula I se technology to check your shlution (s) [8 Blvmpien 1, 22 amd s 7. $$ x^{2}-4 x+3=0 $$ 2. $$ 3 x^{2}+6 x+3=0 $$ 3. $$ x^{2}+a x+15=0 $$ 4. $$ 6 x^{2}-2 x+1=0 $$ S. $$ x^{2}-14 x=-40 $$ 6. $$ 2 x^{2}+4 x=30 $$ $$ 73 x^{2}+5=-2 x $$ 8. $$ -3 x=2 x^{2}-4 $$ 8. $$ -10 x=-25-x^{2} $$ 10. $$ -5 x^{2}-6=-4 x $$ 11. $$ -4 x^{2}+3 x=-5 $$ 12. $$ x^{2}+121=-22 x $$ 13. $$ -z^{2}=-122+6 $$ 14. $$ -7 w+6=-5 w^{2} $$

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UpStudy AI · Accepted Answer

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) \
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

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