quantum tutoring

☝️ Step-by-Step Guide

Alright, let's take a look at what you've got here.

The Problem

The image shows a system of two equations:

y = -1.5
y = x² + 8x + a

It also tells us that a is a positive constant and that this system has exactly one distinct real solution. The question is asking for the value of a.

The Plan

To find the value of a, we can follow these steps:

1. Since both equations are equal to y, we can set the expressions for y equal to each other.
2. Rearrange the resulting equation into the standard form of a quadratic equation, which is Ax² + Bx + C = 0.
3. Recall the condition for a quadratic equation to have exactly one distinct real solution. This involves the discriminant.
4. Use this condition to set up an equation involving a and then solve for a.

How about we start with step 1? What do you get when you set the two expressions for y equal to each other?

💬 Discussion

That's right! You properly set the two expressions for y equal to each other. Now, the next step in our plan is to rearrange this equation, x² + 8x + a = -1.5, into the standard form of a quadratic equation, which is Ax² + Bx + C = 0.

How would you rearrange x² + 8x + a = -1.5 so that one side of the equation is 0?

💬 Discussion

Excellent! You've correctly rearranged the equation into the standard form x² + 8x + (a + 1.5) = 0. Now, we need to use the information that the system has exactly one distinct real solution. This means the quadratic equation we've created has exactly one distinct real solution.

What condition must the discriminant of a quadratic equation (Ax² + Bx + C = 0) satisfy for it to have exactly one distinct real solution?

💡 Concept Exploration

Ah, a great question! Understanding the discriminant is key here.

For a quadratic equation in the standard form Ax² + Bx + C = 0, the discriminant is the expression under the square root in the quadratic formula. It's calculated as:

Discriminant = B² - 4AC

The value of the discriminant tells us about the nature and number of real solutions to the quadratic equation:

• If B² - 4AC > 0, there are two distinct real solutions.
• If B² - 4AC = 0, there is exactly one distinct real solution.
• If B² - 4AC < 0, there are no real solutions (the solutions are complex).

In our equation, x² + 8x + (a + 1.5) = 0, what are the values of A, B, and C?

💬 Discussion

That's exactly right! You've correctly identified the values for A, B, and C in our quadratic equation:

A = 1
B = 8
C = a + 1.5

Now, we know that for the system to have exactly one distinct real solution, the discriminant (B² - 4AC) must be equal to 0.

Using the values you just gave, can you set up the equation for the discriminant and set it equal to 0?

🔍 Doublecheck Work

That's absolutely correct! You've successfully substituted the values of A, B, and C into the discriminant formula and set it equal to 0:

64 - 4(a + 1.5) = 0

This equation is exactly what we need to solve for a.

What's your next step to find the value of a from this equation?

💬 Discussion

Awesome. That's the value I got as well! 🥳

Ready to try another one?