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Algebra

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We are looking to factor $23x^2 + kx - 5.$ Some values of $k$ allow us to factor it as a product of linear binomials with integer coefficients. What are all such values of $k?$
Apr 30, 2023

#1
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For the quadratic to be factorable, there must be 2 numbers that sum to $$k$$ and multiply to $$23 \times -5 =-115$$

There are 4 pairs of numbers that multiply to $$-115$$$$(115, -1)$$$$(23, -5)$$$$(5, -23)$$$$(1, -115)$$

This means that k can be the sum of these values or $$\color{brown}\boxed{k = -114, -18, 18, 114}$$

Apr 30, 2023
#2
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We can factor the quadratic 23x^2 + kx - 5 as a product of linear binomials with integer coefficients if and only if the discriminant of the quadratic is a perfect square. The discriminant is k2−4(23)(−5)=k2+460. This is a perfect square if and only if k2≡−460≡4(mod8). The possible values of k are then k≡±2(mod8), which gives us the solutions k=−2,2,6,−6​.

Apr 30, 2023
#3
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We can factor 23x^2 + kx - 5 as a product of linear binomials with integer coefficients if and only if the discriminant of the quadratic is a perfect square. The discriminant is k2−4⋅23⋅−5=k2+460. This is a perfect square if and only if k2+460=(k+b)2=k2+2bk+b2 for some integer b. This gives us 2bk=460, so b=230. Therefore, k=±(230±460)/2=±345,±115. The possible values of k are −345,−115,115,345​.

Apr 30, 2023
#4
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We can factor 23x^2 + kx - 5 as a product of linear binomials with integer coefficients if and only if the discriminant of the quadratic is a perfect square. The discriminant is k2−4⋅23⋅(−5)=k2+460. This is a perfect square if and only if k2≡0(mod4), which means k≡0,±2,±4,±8(mod10). Therefore, the possible values of k are k∈{0,2,4,8,−2,−4,−8,0}​.

Apr 30, 2023
#5
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The values of k that allow us to factor it as a product of linear binomials with integer coefficients are k ≥ -16√3

What are linear binomials?

Linear binomials are mathematical expression that contain only two terms. Examples include x + 3.

A quadratic equation is an equation in which the highest power of unknown is 2.

How to find the values of k?

Since we have the quadratic equation 23x² + kx - 5 + x² - 3, to determine the value of k which makes it a linear binomial, we simplify and find the discriminant.

What is the discriminant of a quadratic equation?

The discriminant of a quadratic equation ax² + bx + c is D = b² - 4ac

Simplifying the quadratic equation, we have

23x² + kx - 5 + x² - 3 = 23x² + x² + kx - 5 - 3

= 24x² + kx - 8

Comparing the above quadratic equation with ax² + bx + c, we have that

a = 24

b = k and

c = -8

So the discriminant, D =  b² - 4ac

= k² - 4 × 24 × 8

= k² - 768

The allowable values of k are gotten when the discriminant D ≥ 0

So, D ≥ 0

⇒ k² - 768 ≥ 0

⇒ k² ≥ 768

⇒ k ≥ ±√768

⇒ k ≥ ±√(256 × 3)

⇒ k ≥ ±16√3

⇒ k ≥ -16√3

So, the values of k are k ≥ -16√3

May 1, 2023