Use the discriminant to tell whether the
solutions of this equation are real or complex

Question

Use the discriminant to tell whether the solutions of this equation are real or complex $$ 5 x^{2}-7 x+1=0 $$

UpStudy AI · Accepted Answer

To determine whether the solutions of the quadratic equation $$ 5x^2 - 7x + 1 = 0 $$ are real or complex, we can use the discriminant. The discriminant $$ D $$ of a quadratic equation in the form $$ ax^2 + bx + c = 0 $$ is given by the formula:

$$
D = b^2 - 4ac
$$

Here, the coefficients are:
- $$ a = 5 $$
- $$ b = -7 $$
- $$ c = 1 $$

Now, let's calculate the discriminant step by step.

1. Calculate $$ b^2 $$:
   $$
   b^2 = (-7)^2 = 49
   $$

2. Calculate $$ 4ac $$:
   $$
   4ac = 4 \cdot 5 \cdot 1 = 20
   $$

3. Now, substitute these values into the discriminant formula:
   $$
   D = 49 - 20
   $$

4. Calculate $$ D $$:
   $$
   D = 29
   $$

Now, we analyze the value of the discriminant:
- If $$ D > 0 $$, the equation has two distinct real solutions.
- If $$ D = 0 $$, the equation has exactly one real solution (a repeated root).
- If $$ D < 0 $$, the equation has two complex solutions.

Since $$ D = 29 $$ which is greater than 0, we conclude that the solutions of the equation $$ 5x^2 - 7x + 1 = 0 $$ are real and distinct.
The solutions are real and distinct.

Use the discriminant to tell whether the
solutions of this equation are real or complex

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