Find the Values of k for Which k²x² − 2(2k − 1)x + 4 = 0 Has Real and Equal Roots Find the Values of k for Which the Equation Has Real and Equal Roots Solution Given: $$k^2x^2-2(2k-1)x+4=0$$ Here, $$a=k^2,\quad b=-2(2k-1),\quad c=4$$ For real and equal roots, $$D=b^2-4ac=0$$ $$[-2(2k-1)]^2-4(k^2)(4)=0$$ $$4(2k-1)^2-16k^2=0$$ $$ (2k-1)^2-4k^2=0 $$ $$ 4k^2-4k+1-4k^2=0 […]
Find the Value of k for Which the Roots Are Real and Equal in (4 − k)x² + (2k + 4)x + (8k + 1) = 0 Find the Value of k for Which the Roots Are Real and Equal Solution Given: $$ (4-k)x^2+(2k+4)x+(8k+1)=0 $$ Here, $$ a=4-k,\quad b=2k+4,\quad c=8k+1 $$ For real and equal
Find the Values of k for Which x² − 2(k + 1)x + k² = 0 Has Real and Equal Roots Find the Values of k for Which the Equation Has Real and Equal Roots Solution Given: $$x^2-2(k+1)x+k^2=0$$ Here, $$a=1,\quad b=-2(k+1),\quad c=k^2$$ For real and equal roots, $$D=b^2-4ac=0$$ $$[-2(k+1)]^2-4(k^2)=0$$ $$4(k+1)^2-4k^2=0$$ $$ (k+1)^2-k^2=0 $$ $$ k^2+2k+1-k^2=0
Determine the Set of Values of k for Which 3x² + 2x + k = 0 Has Real Roots Determine the Set of Values of k for Which the Equation Has Real Roots Solution Given: $$3x^2+2x+k=0$$ Here, $$a=3,\quad b=2,\quad c=k$$ For real roots, $$D=b^2-4ac\ge0$$ $$2^2-4(3)(k)\ge0$$ $$4-12k\ge0$$ $$k\le\frac{1}{3}$$ Answer The quadratic equation has real roots when
Determine the Set of Values of k for Which kx² + 6x + 1 = 0 Has Real Roots Determine the Set of Values of k for Which the Equation Has Real Roots Solution Given: $$kx^2+6x+1=0$$ Here, $$a=k,\quad b=6,\quad c=1$$ For real roots, $$D=b^2-4ac\ge0$$ $$6^2-4(k)(1)\ge0$$ $$36-4k\ge0$$ $$k\le9$$ Answer The quadratic equation has real roots when
Determine the Set of Values of k for Which 2x² − 5x − k = 0 Has Real Roots Determine the Set of Values of k for Which the Equation Has Real Roots Solution Given: $$2x^2-5x-k=0$$ Here, $$a=2,\quad b=-5,\quad c=-k$$ For real roots, $$D=b^2-4ac\ge0$$ $$(-5)^2-4(2)(-k)\ge0$$ $$25+8k\ge0$$ $$k\ge-\frac{25}{8}$$ Answer The quadratic equation has real roots when
Determine the Set of Values of k for Which 2x² + x + k = 0 Has Real Roots Determine the Set of Values of k for Which the Equation Has Real Roots Solution Given: $$2x^2+x+k=0$$ Here, $$a=2,\quad b=1,\quad c=k$$ For real roots, $$D=b^2-4ac\ge0$$ $$1^2-4(2)(k)\ge0$$ $$1-8k\ge0$$ $$k\le\frac{1}{8}$$ Answer The quadratic equation has real roots when
Determine the Set of Values of k for Which 2x² + 3x + k = 0 Has Real Roots Determine the Set of Values of k for Which the Equation Has Real Roots Solution Given: $$2x^2+3x+k=0$$ Here, $$a=2,\quad b=3,\quad c=k$$ For real roots, $$D=b^2-4ac \ge 0$$ $$3^2-4(2)(k)\ge0$$ $$9-8k\ge0$$ $$k\le\frac{9}{8}$$ Answer The quadratic equation has real
Find the Value of k for Which the Roots Are Real and Equal in 4x² − 2(k + 1)x + (k + 1) = 0 Find the Value of k for Which the Roots Are Real and Equal Solution Given: $$4x^2-2(k+1)x+(k+1)=0$$ Here, $$a=4,\quad b=-2(k+1),\quad c=k+1$$ For real and equal roots, $$D=b^2-4ac=0$$ $$[-2(k+1)]^2-4(4)(k+1)=0$$ $$4(k+1)^2-16(k+1)=0$$ $$ (k+1)\big[(k+1)-4\big]=0
Find the Value of k for Which the Roots Are Real and Equal in 4x² − 2(k + 1)x + (k + 4) = 0 Find the Value of k for Which the Roots Are Real and Equal Solution Given: $$4x^2-2(k+1)x+(k+4)=0$$ Here, $$a=4,\quad b=-2(k+1),\quad c=k+4$$ For real and equal roots, $$D=b^2-4ac=0$$ $$[-2(k+1)]^2-4(4)(k+4)=0$$ $$4(k+1)^2-16(k+4)=0$$ $$ (k+1)^2-4(k+4)=0