400y_1^2+2×7.54×(1017.87+1017.87) y_1-2×7.54×(1017.87×430+1017.87×70)=0

Let's break down the given equation and solve for \( y_1 \).

The equation is:

\[
400 y_1^2 + 2 \times 7.54 \times (1017.87 + 1017.87) y_1 - 2 \times 7.54 \times (1017.87 \times 430 + 1017.87 \times 70) = 0
\]

### Step 1: Simplify the equation

#### Simplifying the coefficients:
- \( 1017.87 + 1017.87 = 2035.74 \)
- \( 1017.87 \times 430 = 437684.1 \)
- \( 1017.87 \times 70 = 71250.9 \)
- \( 437684.1 + 71250.9 = 508935 \)

Now, substitute these values back into the equation:

\[
400 y_1^2 + 2 \times 7.54 \times 2035.74 y_1 - 2 \times 7.54 \times 508935 = 0
\]

#### Calculate the constant terms:
- \( 2 \times 7.54 = 15.08 \)
- \( 15.08 \times 2035.74 = 30681.8 \)
- \( 15.08 \times 508935 = 7665601.8 \)

Now the equation becomes:

\[
400 y_1^2 + 30681.8 y_1 - 7665601.8 = 0
\]

### Step 2: Solve using the quadratic formula

The quadratic formula is:

\[
y_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 400 \), \( b = 30681.8 \), and \( c = -7665601.8 \). Let's compute the discriminant \( \Delta = b^2 - 4ac \) first:

\[
\Delta = 30681.8^2 - 4 \times 400 \times (-7665601.8)
\]

Let me calculate the discriminant and then solve for \( y_1 \).The discriminant \( \Delta = 13,206,335,731.24 \), which is a positive value. Since the discriminant is positive, there will be two real solutions for \( y_1 \).

Let's now compute the two possible values for \( y_1 \) using the quadratic formula:

\[
y_1 = \frac{-b \pm \sqrt{\Delta}}{2a}
\]

I'll compute both solutions.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can still proceed with the formula manually. Here's how you can calculate the two possible values for \( y_1 \):

\[
y_1 = \frac{-30681.8 \pm \sqrt{13206335731.24}}{2 \times 400}
\]

Just calculate \( \sqrt{13206335731.24} \) and then use both the plus and minus signs in the formula to get the two solutions for \( y_1 \).

If you have any other questions or need further details, feel free to ask!

### Here are five related questions you might find interesting:
1. How does the discriminant determine the nature of the roots in quadratic equations?
2. What are the steps involved in completing the square for solving quadratic equations?
3. Can all quadratic equations be factored, and if not, why?
4. How is the quadratic formula derived from completing the square?
5. What are some practical applications of quadratic equations in physics and engineering?

#### Tip:
Always check the sign of the discriminant before solving quadratic equations. It tells you whether the solutions are real or complex.

400y_1^2+2×7.54×(1017.87+1017.87)y_1-2×7.54×(1017.87×430+1017.87×70)=0

Learn how to solve the quadratic equation 400y_1^2 + 30681.8y_1 - 7665601.8 = 0 with a step-by-step solution. This problem involves calculating the discriminant and applying the quadratic formula. Suitable for students in Grades 9-12.

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