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Answer :
We start with the quadratic equation
[tex]$$-8x^2 - 13x = 0.$$[/tex]
Step 1: Identify the coefficients.
The standard form of a quadratic equation is
[tex]$$ax^2 + bx + c = 0.$$[/tex]
Here, we have
[tex]$$a = -8,\quad b = -13, \quad c = 0.$$[/tex]
Step 2: Compute the discriminant.
The discriminant of a quadratic equation is given by
[tex]$$D = b^2 - 4ac.$$[/tex]
Substitute the coefficients:
[tex]$$
D = (-13)^2 - 4(-8)(0) = 169 - 0 = 169.
$$[/tex]
Step 3: Determine the number and type of solutions.
Since the discriminant is positive ([tex]$D = 169 > 0$[/tex]), the quadratic equation has two distinct real solutions. Given that the equation factors easily as
[tex]$$x(-8x - 13) = 0,$$[/tex]
the solutions are
[tex]$$x = 0 \quad \text{and} \quad x = -\frac{13}{8}.$$[/tex]
Both solutions are rational.
Final Answer:
The discriminant is [tex]$169$[/tex]; therefore, the quadratic has two rational solutions.
[tex]$$-8x^2 - 13x = 0.$$[/tex]
Step 1: Identify the coefficients.
The standard form of a quadratic equation is
[tex]$$ax^2 + bx + c = 0.$$[/tex]
Here, we have
[tex]$$a = -8,\quad b = -13, \quad c = 0.$$[/tex]
Step 2: Compute the discriminant.
The discriminant of a quadratic equation is given by
[tex]$$D = b^2 - 4ac.$$[/tex]
Substitute the coefficients:
[tex]$$
D = (-13)^2 - 4(-8)(0) = 169 - 0 = 169.
$$[/tex]
Step 3: Determine the number and type of solutions.
Since the discriminant is positive ([tex]$D = 169 > 0$[/tex]), the quadratic equation has two distinct real solutions. Given that the equation factors easily as
[tex]$$x(-8x - 13) = 0,$$[/tex]
the solutions are
[tex]$$x = 0 \quad \text{and} \quad x = -\frac{13}{8}.$$[/tex]
Both solutions are rational.
Final Answer:
The discriminant is [tex]$169$[/tex]; therefore, the quadratic has two rational solutions.
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