Answer:
The quadratic equation in standard form is [tex]y= x^{2}-\frac{7}{4}\cdot x +\frac{3}{8}[/tex].
Explanation:
From Algebra, we remember that the order of the polynomial indicates the number of roots contained within. The factorized form of a second order polynomial (quadratic equation) is now introduced:
[tex]y = (x-r_{1})\cdot (x-r_{2})[/tex] (1)
Where:
[tex]x[/tex] - Independent variable.
[tex]y[/tex] - Dependent variable.
[tex]r_{1}[/tex], [tex]r_{2}[/tex] - Roots of the polynomial.
And the standard form of the quadratic equation:
[tex]y = a\cdot x^{2}+b\cdot x +c[/tex] (2)
Where [tex]a[/tex], [tex]b[/tex], [tex]c[/tex] are the coefficients of the polynomial.
Lastly, we proceed to perform algebraic operations until standard form is found:
1) [tex]r_{1} = \frac{3}{2}[/tex], [tex]r_{2} = \frac{1}{4}[/tex] Given
2) [tex]y = \left(x-\frac{3}{2}\right)\cdot \left(x-\frac{1}{4} \right)[/tex] By 1)/Definition of factorized form
3) [tex]y = \left[x+\left(-\frac{3}{2} \right)\right]\cdot \left[x+\left(-\frac{1}{4} \right)\right][/tex] Definition of subtraction
4) [tex]y = \left[x+\left(-\frac{3}{2} \right)\right]\cdot x +\left[x+\left(-\frac{3}{2} \right)\right]\cdot \left(-\frac{1}{4} \right)[/tex] Distributive property
5) [tex]y = x\cdot x +\left(-\frac{3}{2} \right)\cdot x +\left(-\frac{1}{4}\right)\cdot x +\left(-\frac{3}{2}\right)\cdot \left(-\frac{1}{4} \right)[/tex] Distributive and commutative properties
6) [tex]y = x^{2} + \left[-\left(\frac{3}{2}+\frac{1}{4}\right)\right]\cdot x +\left(-\frac{3}{2} \right)\cdot \left(-\frac{1}{4} \right)[/tex] Distributive and associative properties/[tex](-1)\cdot a = -a[/tex]
7) [tex]y = x^{2}+\left(-\frac{7}{4} \right)\cdot x + \frac{3}{8}[/tex] [tex]\frac{a}{b}\times \frac{c}{d} = \frac{a\cdot c}{b\cdot d}[/tex]/[tex](-a)\cdot (-b) = a\cdot b[/tex]
8) [tex]y= x^{2}-\frac{7}{4}\cdot x +\frac{3}{8}[/tex] Definition of substraction/Result
