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Place the following equations to its corresponding pathway

To solve the given quadratic equations and convert them to standard form \( ax^2 + bx + c = 0 \), we can follow these steps:

1. **Given Equation**: \(4x^2 - 11x = 7\)
   - Subtract 7 from both sides to set the equation to zero:
     \[
     4x^2 - 11x - 7 = 0
     \]
   - The standard form is \(4x^2 - 11x - 7 = 0\), with \(a = 4\), \(b = -11\), and \(c = -7\).

2. **Given Equation**: \(5x^2 = 3x\)
   - Subtract \(3x\) from both sides to set the equation to zero:
     \[
     5x^2 - 3x = 0
     \]
   - The standard form is \(5x^2 - 3x = 0\), with \(a = 5\), \(b = -3\), and \(c = 0\).

3. **Given Equation**: \((x + 3)(x - 2) = 1\)
   - Expand the left-hand side:
     \[
     x^2 - 2x + 3x - 6 = x^2 + x - 6
     \]
   - Subtract 1 from both sides to set the equation to zero:
     \[
     x^2 + x - 7 = 0
     \]
   - The standard form is \(x^2 + x - 7 = 0\), with \(a = 1\), \(b = 1\), and \(c = -7\).

4. **Given Equation**: \((x - 7)^2 + 3 = 0\)
   - Expand \((x - 7)^2\):
     \[
     (x - 7)^2 = x^2 - 14x + 49
     \]
   - Substitute back into the equation:
     \[
     x^2 - 14x + 49 + 3 = x^2 - 14x + 52 = 0
     \]
   - The standard form is \(x^2 - 14x + 52 = 0\), with \(a = 1\), \(b = -14\), and \(c = 52\).

5. **Given Equation**: \(6x^2 = 5x - 4\)
   - Subtract \(5x - 4\) from both sides to set the equation to zero:
     \[
     6x^2 - 5x + 4 = 0
     \]
   - The standard form is \(6x^2 - 5x + 4 = 0\), with \(a = 6\), \(b = -5\), and \(c = 4\).

Here is the completed table:

| GIVEN                   | STANDARD FORM          | a  | b   | c   |
|-------------------------|------------------------|----|-----|-----|
| \(4x^2 - 11x = 7\)      | \(4x^2 - 11x - 7 = 0\) | 4  | -11 | -7  |
| \(5x^2 = 3x\)           | \(5x^2 - 3x = 0\)      | 5  | -3  | 0   |
| \((x + 3)(x - 2) = 1\)  | \(x^2 + x - 7 = 0\)    | 1  | 1   | -7  |
| \((x - 7)^2 + 3 = 0\)   | \(x^2 - 14x + 52 = 0\) | 1  | -14 | 52  |
| \(6x^2 = 5x - 4\)       | \(6x^2 - 5x + 4 = 0\)  | 6  | -5  | 4   |

Would you like any further details or have any questions?

Here are 8 related questions:
1. How do you factor quadratic equations?
2. What methods can be used to solve quadratic equations?
3. How does the discriminant of a quadratic equation determine the number of solutions?
4. Can you give an example of solving a quadratic equation by completing the square?
5. How do you graph a quadratic equation?
6. What is the significance of the vertex in a quadratic equation?
7. How do you determine the axis of symmetry for a quadratic function?
8. How can quadratic equations be applied in real-life scenarios?

**Tip**: Always ensure to set the quadratic equation to zero before identifying the coefficients \(a\), \(b\), and \(c\).

Explore how to solve quadratic equations with detailed steps and examples. This page covers converting quadratic equations to standard form, identifying coefficients, and applying factorization methods. Suitable for Grades 9-12.

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