Answer:

[tex]\boxed{\bf\:x = \dfrac{5}{4} \: , \: - \: 2 \: }\\ [/tex]

Explanation:

Given quadratic equation is

[tex]\bf \: {4x}^{2} + 3x - 10 = 0 \\ [/tex]

On comparing with general equation ax² + bx + c = 0, we get

[tex] \:\boxed{\begin{aligned}& \qquad \:\sf \: a=4 \qquad \: \\ \\& \qquad \:\sf \: b=3\\ \\& \qquad \:\sf \: c=- 10\end{aligned}} \qquad \:\\ [/tex]

So, Solution of quadratic equation using quadratic formula is

[tex]\sf \: x = \dfrac{ - b \: \pm \: \sqrt{ {b}^{2} - 4ac} }{2a} \\ [/tex]

On substituting the values of a, b and c, we get

[tex]\sf \: x = \dfrac{ - 3\: \pm \: \sqrt{ {3}^{2} - 4(4)(- 10)} }{2(4)} \\ [/tex]

[tex]\sf \: x = \dfrac{ - 3\: \pm \: \sqrt{9 + 160} }{8} \\ [/tex]

[tex]\sf \: x = \dfrac{ - 3\: \pm \: \sqrt{169} }{8} \\ [/tex]

[tex]\sf \: x = \dfrac{ - 3\: \pm \: 13 }{8} \\ [/tex]

[tex]\sf \: x = \dfrac{ - 3 \:+\:13 }{8} \: \: or \: \: \dfrac{ - 3 \: - \:13 }{8}\\ [/tex]

[tex]\sf\: x = \dfrac{ 10 }{8} \: \: or \: \: \dfrac{ - 16 }{8}\\ [/tex]

[tex]\implies\sf\: x = \dfrac{5}{4} \: \: or \: \: - 2\\ [/tex]

Hence,

[tex]\implies\sf\: \boxed{\bf\:x = \dfrac{5}{4} \: , \: - \: 2 \: }\\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Additional Information:

Nature of roots

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where, Discriminant, D = b² - 4ac