Solve the Quadratic Equation
From Fist Pinciple.

to

Ask by Gibbs Hill. in Australia

Dec 27,2024

Question

Solve the Quadratic Equation From Fist Pinciple. $$ y=a x^{2}+b x+c $$ to $$ y=\frac{-b \pm \sqrt{4 a c}-b^{2}}{29} $$

UpStudy AI · Accepted Answer

Certainly! Let's derive the **Quadratic Formula** from the **first principles** for the general quadratic equation:

$$
ax^2 + bx + c = 0
$$

where $$ a \neq 0 $$.

### Step 1: Start with the Standard Form

Begin with the standard quadratic equation:

$$
ax^2 + bx + c = 0
$$

### Step 2: Divide Both Sides by $$ a $$

To simplify, divide every term by $$ a $$ (since $$ a \neq 0 $$):

$$
x^2 + \frac{b}{a}x + \frac{c}{a} = 0
$$

### Step 3: Move the Constant Term to the Right Side

$$
x^2 + \frac{b}{a}x = -\frac{c}{a}
$$

### Step 4: Complete the Square

To complete the square, take half of the coefficient of $$ x $$, square it, and add it to both sides.

1. **Find half of $$ \frac{b}{a} $$:**

$$
\frac{b}{2a}
$$

2. **Square it:**

$$
\left( \frac{b}{2a} \right)^2 = \frac{b^2}{4a^2}
$$

3. **Add this square to both sides:**

$$
x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}
$$

### Step 5: Rewrite the Left Side as a Perfect Square

The left side now forms a perfect square:

$$
\left( x + \frac{b}{2a} \right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}
$$

### Step 6: Simplify the Right Side

Combine the terms on the right side:

$$
\left( x + \frac{b}{2a} \right)^2 = \frac{-4ac + b^2}{4a^2}
$$

$$
\left( x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2}
$$

### Step 7: Take the Square Root of Both Sides

$$
x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}
$$

### Step 8: Solve for $$ x $$

Subtract $$ \frac{b}{2a} $$ from both sides:

$$
x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}
$$

Combine the terms over a common denominator:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

### **Quadratic Formula**

Thus, the solutions to the quadratic equation $$ ax^2 + bx + c = 0 $$ are given by:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

### **Addressing the Provided Expression**

You mentioned the expression:

$$
y = \frac{-b \pm \sqrt{4ac} - b^2}{29}
$$

It seems there might be a misunderstanding in the formula you've provided. Let's clarify the correct components:

1. **Discriminant:** The term under the square root in the quadratic formula is $$ b^2 - 4ac $$, not $$ 4ac - b^2 $$ or $$ \sqrt{4ac} - b^2 $$.

2. **Denominator:** The denominator in the quadratic formula is $$ 2a $$, not 29.

3. **Overall Structure:** The correct quadratic formula has the square root applied solely to the discriminant.

### **Correct Formula Comparison**

$$
\text{Your Expression: } y = \frac{-b \pm \sqrt{4ac} - b^2}{29}
$$

$$
\text{Correct Quadratic Formula: } x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

Ensure that:

- The discriminant is $$ b^2 - 4ac $$.
- The entire numerator (including the square root term) is over $$ 2a $$.

If you have specific values for $$ a $$, $$ b $$, and $$ c $$, feel free to share them, and I can help compute the solutions using the correct quadratic formula!
The quadratic formula for the equation $$ ax^2 + bx + c = 0 $$ is: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ This formula provides the solutions for $$ x $$ in the quadratic equation.

Solve the Quadratic Equation
From Fist Pinciple.

to

Upstudy AI Solution

Answer

Solution

Step 1: Start with the Standard Form

Step 2: Divide Both Sides by

Step 3: Move the Constant Term to the Right Side

Step 4: Complete the Square

Step 5: Rewrite the Left Side as a Perfect Square

Step 6: Simplify the Right Side

Step 7: Take the Square Root of Both Sides

Step 8: Solve for

Quadratic Formula

Addressing the Provided Expression

Correct Formula Comparison

The Deep Dive

Related Questions

Latest Algebra Questions

Solve the Quadratic Equation From Fist Pinciple. to