*determine the number and type of solutions to the Quadratic Equation x2-6x=7*


To determine the number and type of solutions to the quadratic equation \(x^2 - 6x = 7\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). We need to move the constant term to the left-hand side:

\[
x^2 - 6x - 7 = 0
\]

Now, the equation is in standard form, where \(a = 1\), \(b = -6\), and \(c = -7\).

### Step 2: Use the discriminant to determine the number and type of solutions
The discriminant, \(D\), of a quadratic equation \(ax^2 + bx + c = 0\) is given by:

\[
D = b^2 - 4ac
\]

Substitute the values of \(a\), \(b\), and \(c\):

\[
D = (-6)^2 - 4(1)(-7) = 36 + 28 = 64
\]

### Step 3: Interpret the discriminant
- If \(D > 0\), the equation has two real and distinct solutions.
- If \(D = 0\), the equation has one real repeated solution.
- If \(D < 0\), the equation has two complex (non-real) solutions.

Since \(D = 64\), which is greater than 0, the quadratic equation has **two real and distinct solutions**.

### Step 4: Solve the equation (optional)
If you want to find the actual solutions, you can apply the quadratic formula:

\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]

Substitute \(b = -6\), \(a = 1\), and \(D = 64\):

\[
x = \frac{-(-6) \pm \sqrt{64}}{2(1)} = \frac{6 \pm 8}{2}
\]

So, the two solutions are:

\[
x_1 = \frac{6 + 8}{2} = \frac{14}{2} = 7
\]
\[
x_2 = \frac{6 - 8}{2} = \frac{-2}{2} = -1
\]

### Conclusion:
The quadratic equation \(x^2 - 6x = 7\) has two real and distinct solutions: \(x = 7\) and \(x = -1\).

Would you like more details or have any questions?

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Here are 5 related questions:
1. What is the discriminant and how is it used to classify solutions?
2. How do you solve a quadratic equation by factoring?
3. Can you explain how the quadratic formula is derived?
4. How can you complete the square for solving quadratic equations?
5. What is the graphical interpretation of the solutions of a quadratic equation?

**Tip:** For quicker solutions, always check if the quadratic equation can be factored before using the quadratic formula!

determine the number and type of solutions to the Quadratic Equation x^2 - 6x = 7

Learn how to solve the quadratic equation x^2 - 6x = 7 step-by-step. This guide covers core algebra concepts such as the quadratic formula, discriminant, and how to classify the solutions. Ideal for students in Grades 8-10.

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