Question

Can someone help me with these two problems and show work please !!

Answer 1

work is attached to this post

Answer 2

Answer:

13. Two real solutions.

14. No real solutions.

Step-by-step explanation:

Given quadratic equations:

13. $x^2+7x+10=0$

14. $4x^2-3x+4=0$

${\large \textsf{Quadratic Formula: }} x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}\ \Bigg{\|}\ \textsf{when }ax^2+bx+c=0$

What is the discriminant?

The discriminant (Δ) is b² - 4ac, which is under the square root in the quadratic formula. It is used to determine the number of real solutions (roots) of a quadratic:

• When b² - 4ac > 0, the quadratic has two real solutions.
• When b² - 4ac = 0, the quadratic has one real solution.
• When b² - 4ac < 0, the quadratic has no real (complex) solutions.

13. x² + 7x + 10 = 0

$\implies a=\textsf{1},b=\textsf{7},c=\textsf{10}$

Find the discriminant value.

$\implies \Delta = b^2 - 4ac$

$\implies \Delta=(7)^2 - 4(1)(10)$

$\implies \Delta=49 - 40$

$\implies \Delta=9$

As $9$ > 0, this quadratic has two real solutions.

14. 4x² - 3x + 4 = 0

$\implies a=\textsf{4},b=\textsf{-3},c=\textsf{4}$

Find the discriminant value.

$\implies \Delta = b^2 - 4ac$

$\implies \Delta = (-3)^2 - 4(4)(4)$

$\implies \Delta = 9 - 64$

$\implies \Delta = -55$

As $-55$ < 0, this quadratic has no real solutions.

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